Advanced Math Alıştırma Soruları — SAT — Sayfa 12

Soru 221Soru →

A polynomial $p$ is defined by $p(x) = (x - 4)(x^2 + ax + 3)$ , where $a$ is a constant. If $p(1) = -18$ , what is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2

Substituting

x = 1

into

p(x) = (x - 4)(x^2 + ax + 3)

gives

p(1) = (1 - 4)(1^2 + a(1) + 3) = -3(4 + a)

. Since we are given that

p(1) = -18

, we can set

-3(4 + a) = -18

. Dividing both sides of the equation by

-3

gives

4 + a = 6

. Subtracting

4

from both sides gives

a = 2

.

Adım Adım Çözüm

1

Substitute

x = 1

into the expression for

p(x)

.

p(1) = (1 - 4)(1^2 + a(1) + 3) = -3(4 + a)

We are given that

p(1) = -18

, so substituting

x = 1

will allow us to set up an equation in terms of the constant

a

.

2

Set the simplified expression equal to

-18

and solve for

a

.

a = 2

Dividing both sides of

-3(4 + a) = -18

by

-3

yields

4 + a = 6

, and subtracting

4

from both sides gives the value of

a

.

Anahtar Kavram

Using given function values to solve for unknown coefficients in a factored polynomial.

Soru 222Soru →

In the $xy$ -plane, the circle with equation $x^2 + y^2 = 13$ intersects the line with equation $y - x = 1$ at two points, $(x_1, y_1)$ and $(x_2, y_2)$ . What is the value of $y_1 + y_2$ ?

-1

1

3

-5

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The sum of the

y

-coordinates of the intersection points is

1

.

The correct answer is

1

. Substituting

x = y - 1

from the linear equation into the circle equation yields

(y-1)^2 + y^2 = 13

. Expanding and simplifying gives the quadratic equation

2y^2 - 2y - 12 = 0

. Dividing by 2 yields

y^2 - y - 6 = 0

, which factors into

(y - 3)(y + 2) = 0

. Therefore, the

y

-coordinates of the intersection points are

3

and

-2

. The sum of these coordinates is

3 + (-2) = 1

.

Adım Adım Çözüm

1

Rearrange the linear equation to solve for

x

in terms of

y

.

x = y - 1

Expressing

x

in terms of

y

allows us to substitute it into the circle equation to find the

y

-coordinates directly.

2

Substitute

x = y - 1

into the circle equation

x^2 + y^2 = 13

.

(y - 1)^2 + y^2 = 13

This substitution reduces the system of equations to a single quadratic equation in terms of

y

.

3

Expand and simplify the quadratic equation into standard form

ay^2 + by + c = 0

.

y^2 - 2y + 1 + y^2 = 13 \implies 2y^2 - 2y - 12 = 0

Expanding the squared term and combining like terms puts the equation in a form that can be solved.

4

Divide the equation by 2 and factor the simplified quadratic equation.

y^2 - y - 6 = 0 \implies (y - 3)(y + 2) = 0

Simplifying and factoring helps find the individual

y

-coordinates of the two intersection points.

5

Solve for

y

and calculate the sum of the coordinates.

y = 3

or

y = -2

, and

3 + (-2) = 1

Solving the factored equation gives the

y

-coordinates of the two intersection points, which are then added to find the final value.

Anahtar Kavram

Solving a nonlinear system of equations representing a circle and a line via substitution.

Alternatif Yöntem

Instead of solving for

x

first, we can substitute

y = x + 1

into the circle equation to find the

x

-coordinates:

x^2 + (x + 1)^2 = 13 \implies 2x^2 + 2x - 12 = 0 \implies x^2 + x - 6 = 0 \implies (x + 3)(x - 2) = 0

. This gives

x = -3

or

x = 2

. We then find the corresponding

y

-values using

y = x + 1

: for

x = -3

,

y = -2

; for

x = 2

,

y = 3

. The sum of the

y

-coordinates is

-2 + 3 = 1

.

Tahmini Süre:1m 30s

Soru 223Soru →

If $x$ is the real solution to the equation $\sqrt{5x + 6} - \sqrt{x + 3} = 3$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

The correct answer is

6

. Isolating the first radical gives

\sqrt{5x+6} = \sqrt{x+3} + 3

. Squaring both sides yields

5x+6 = x+12+6\sqrt{x+3}

, which simplifies to

4x-6 = 6\sqrt{x+3}

. Dividing by

2

and squaring both sides again gives

(2x-3)^2 = 9(x+3)

, leading to the quadratic equation

4x^2 - 21x - 18 = 0

. Factoring this equation yields

(4x+3)(x-6)=0

, which gives the potential solutions

x = 6

and

x = -\frac{3}{4}

. Checking both solutions in the original equation shows that

x = 6

is a valid solution because

\sqrt{36} - \sqrt{9} = 6 - 3 = 3

. The solution

x = -\frac{3}{4}

is extraneous because

\sqrt{2.25} - \sqrt{2.25} = 0 \neq 3

.

Adım Adım Çözüm

1

Isolate the first radical term and square both sides.

5x + 6 = x + 12 + 6\sqrt{x + 3}

Isolating one radical term simplifies the squaring process to eliminate that radical.

2

Isolate the remaining radical term and simplify the expression.

2x - 3 = 3\sqrt{x + 3}

Simplifying and isolating the remaining radical term prepares the equation for a second squaring step.

3

Square both sides to eliminate the remaining radical and form a quadratic equation.

4x^2 - 21x - 18 = 0

Squaring both sides removes the last radical, turning the radical equation into a polynomial equation.

4

Factor the quadratic equation to find all candidate solutions.

x = 6

or

x = -\frac{3}{4}

Factoring

(4x + 3)(x - 6) = 0

gives the roots of the quadratic equation.

5

Substitute candidate solutions back into the original equation to identify extraneous roots.

The only valid solution is

x = 6

; the solution

x = -\frac{3}{4}

is extraneous.

Squaring equations can introduce extraneous solutions, so candidate solutions must be verified in the original equation.

Anahtar Kavram

Solving equations with multiple radicals by isolating terms, squaring both sides, and verifying candidate solutions for extraneous roots.

Soru 224Soru →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -(x - d)^2 + d^2$ , where $d$ is a positive constant, has vertex $V$ and intersects the $x$ -axis at points $P$ and $Q$ . If the area of triangle $PVQ$ is $64$ , what is the value of $d$ ?

2

4

8

16

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

d

is

4

.

The function

f(x) = -(x - d)^2 + d^2

is in vertex form,

f(x) = a(x-h)^2 + k

, so its vertex is

V(d, d^2)

. Since

d > 0

, the vertex is in the first quadrant, and the height of the triangle is

d^2

. Setting

f(x) = 0

gives the

x

-intercepts

P(0,0)

and

Q(2d,0)

, meaning the base of the triangle has a length of

2d

. Using the area of a triangle formula, the area is

\frac{1}{2} \times 2d \times d^2 = d^3

. Since the area is given as

64

, we set

d^3 = 64

, which yields

d = 4

.

Adım Adım Çözüm

1

Identify the vertex

V

of the quadratic function

f(x) = -(x - d)^2 + d^2

.

The vertex is

V(d, d^2)

.

The function is written in vertex form,

f(x) = a(x - h)^2 + k

, where the vertex is

(h, k)

.

2

Find the

x

-intercepts

P

and

Q

of the function by setting

f(x) = 0

.

x = 0

and

x = 2d

.

Setting

-(x - d)^2 + d^2 = 0

gives

(x - d)^2 = d^2

, which simplifies to

x - d = \pm d

.

3

Calculate the area of triangle

PVQ

in terms of

d

.

The area is

d^3

.

The base of the triangle along the

x

-axis is the distance between the intercepts,

2d - 0 = 2d

. The height is the

y

-coordinate of the vertex,

d^2

. The area is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} (2d)(d^2) = d^3

.

4

Solve for

d

using the given area of

64

.

d = 4

.

Setting the area expression

d^3

equal to

64

and taking the cube root of both sides gives

d = 4

.

Anahtar Kavram

Finding the vertex and intercepts of a quadratic function in vertex form and applying geometric formulas to analyze the graph.

Soru 225Soru →

What is the sum of all real solutions to the equation $\frac{x - 5}{x - 1} = \frac{2}{x + 1} - \frac{8}{x^2 - 1}$ ?

1

5

6

-6

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The sum of all real solutions to the equation is

5

.

Multiplying the equation by the least common denominator,

(x - 1)(x + 1)

, yields

(x - 5)(x + 1) = 2(x - 1) - 8

. Expanding and simplifying leads to the quadratic equation

x^2 - 6x + 5 = 0

, which factors as

(x - 1)(x - 5) = 0

. This gives potential solutions of

x = 1

and

x = 5

. Since

x = 1

makes the denominators of the original terms zero, it is extraneous and must be discarded. The only valid solution is

x = 5

, and therefore the sum of all real solutions is

5

.

Adım Adım Çözüm

1

Find the least common denominator of the rational terms.

The denominators are

x - 1

,

x + 1

, and

x^2 - 1

. Since

x^2 - 1 = (x - 1)(x + 1)

, the least common denominator is

(x - 1)(x + 1)

.

Finding the least common denominator allows us to eliminate the fractions by multiplying both sides.

2

Multiply the entire equation by the least common denominator

(x - 1)(x + 1)

, assuming

x \neq 1

and

x \neq -1

.

(x - 5)(x + 1) = 2(x - 1) - 8

This clears all rational expressions, leaving a polynomial equation.

3

Expand both sides of the equation and combine like terms.

x^2 - 4x - 5 = 2x - 2 - 8 \Rightarrow x^2 - 4x - 5 = 2x - 10

Expanding allows the simplification of terms on each side of the equation.

4

Rearrange the equation to set it equal to zero.

x^2 - 6x + 5 = 0

Moving all terms to one side forms a standard quadratic equation which can then be solved.

5

Factor the quadratic equation.

(x - 1)(x - 5) = 0

, which gives potential solutions

x = 1

and

x = 5

.

Factoring allows us to find the roots of the quadratic equation.

6

Check the potential solutions in the original equation to identify any extraneous solutions.

Substituting

x = 1

results in division by zero in the terms

\frac{x - 5}{x - 1}

and

\frac{8}{x^2 - 1}

, so

x = 1

is extraneous. Substituting

x = 5

yields

0 = 0

, meaning

x = 5

is a valid real solution.

Multiplying by variables can introduce extraneous solutions that make the original denominators zero.

7

Calculate the sum of all valid solutions.

Since

x = 5

is the only valid solution, the sum of all real solutions is

5

.

The question asks for the sum of all real solutions.

Anahtar Kavram

Solving rational equations by clearing denominators and checking for extraneous solutions.

Soru 226Soru →

If $4^x - 3^{x - 0.5} = 3^{x + 0.5} - 2^{2x-1}$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 1.5

Cevap

The correct answer is

1.5

(or

\frac{3}{2}

)

By rewriting

4^x

as

2^{2x}

and grouping the base 2 and base 3 terms, the equation becomes

2^{2x} + 2^{2x-1} = 3^{x+0.5} + 3^{x-0.5}

. Factoring out the variable terms gives

2^{2x}(1 + 2^{-1}) = 3^x(3^{0.5} + 3^{-0.5})

. Simplifying the coefficients results in

\frac{3}{2} \cdot 2^{2x} = \frac{4}{\sqrt{3}} \cdot 3^x

. Dividing both sides to separate the variables and constants yields

\frac{2^{2x}}{3^x} = \frac{8}{3\sqrt{3}}

. Representing both sides with the common base

\frac{4}{3}

gives

\left(\frac{4}{3}\right)^x = \left(\frac{4}{3}\right)^{1.5}

. Equating the exponents leads to

x = 1.5

.

Adım Adım Çözüm

1

Rewrite the base 4 term as a power of 2

4^x = (2^2)^x = 2^{2x}

Expressing exponential terms with prime bases simplifies finding relationships between them.

2

Group terms with like bases on opposite sides of the equation

2^{2x} + 2^{2x-1} = 3^{x+0.5} + 3^{x-0.5}

Separating different base systems allows us to factor out common exponential functions.

3

Factor out the common exponential factors from each side

2^{2x}(1 + 2^{-1}) = 3^x(3^{0.5} + 3^{-0.5})

This isolates the constant coefficients from the exponential functions of

x

.

4

Simplify the constant terms on both sides of the equation

2^{2x}\left(\frac{3}{2}\right) = 3^x\left(\sqrt{3} + \frac{1}{\sqrt{3}}\right) = 3^x\left(\frac{4}{\sqrt{3}}\right)

Reducing the coefficients makes it easier to combine the constant terms.

5

Divide both sides to group the exponential terms together

\frac{2^{2x}}{3^x} = \frac{4}{\sqrt{3}} \cdot \frac{2}{3} = \frac{8}{3\sqrt{3}}

This sets up the equation in the form of

f(x) = C

where

f(x)

has a combined base.

6

Express both sides of the equation with a common base of

\frac{4}{3}

\left(\frac{4}{3}\right)^x = \left(\frac{4}{3}\right)^{1.5}

Since

2^{2x} = (2^2)^x = 4^x

, the left side simplifies to

\left(\frac{4}{3}\right)^x

. The right side can be rewritten as

\frac{8}{3\sqrt{3}} = \frac{2^3}{3^{1.5}} = \frac{(4^{0.5})^3}{3^{1.5}} = \frac{4^{1.5}}{3^{1.5}} = \left(\frac{4}{3}\right)^{1.5}

.

7

Equate the exponents

x = 1.5

Since the bases are identical and positive, the exponents must be equal.

Anahtar Kavram

Solving exponential equations using base conversion, exponent rules, and factoring.

Soru 227Soru →

If $\frac{2}{x} + \frac{3}{x - 2} = 2$ and $x > 1$ , what is the value of $x - 1$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The correct answer is 3.

By multiplying both sides of the equation by the common denominator

x(x - 2)

, we obtain the quadratic equation

2x^2 - 9x + 4 = 0

. Factoring this equation yields the solutions

x = 0.5

and

x = 4

. Since the problem specifies that

x > 1

, we select

x = 4

. Substituting this value into the expression

x - 1

gives the final answer of 3.

Adım Adım Çözüm

1

Multiply the equation by the common denominator

x(x - 2)

to eliminate the denominators.

2(x - 2) + 3x = 2x(x - 2)

This clears the fractions and allows us to work with a polynomial equation.

2

Expand and simplify both sides of the equation.

5x - 4 = 2x^2 - 4x

Combining like terms simplifies the expression before solving.

3

Rearrange the terms to set the quadratic equation equal to zero.

2x^2 - 9x + 4 = 0

Putting the equation in standard form

ax^2 + bx + c = 0

is necessary for factoring.

4

Factor the quadratic expression.

(2x - 1)(x - 4) = 0

Factoring allows us to find the roots of the quadratic equation.

5

Solve for

x

and apply the condition

x > 1

.

x = 4

The equation yields two solutions:

x = 0.5

and

x = 4

. The condition

x > 1

excludes

x = 0.5

, leaving

x = 4

as the only valid solution.

6

Calculate the value of the requested expression

x - 1

.

3

Substituting

x = 4

into

x - 1

gives

4 - 1 = 3

.

Anahtar Kavram

Solving rational equations by clearing denominators and solving the resulting quadratic equation while considering domain constraints.

Alternatif Yöntem

Instead of factoring, the quadratic formula can be used to solve

2x^2 - 9x + 4 = 0

, where

x = \frac{9 \pm \sqrt{(-9)^2 - 4(2)(4)}}{2(2)} = \frac{9 \pm 5}{4}

, giving

x = 0.5

and

x = 4

. Applying

x > 1

leaves

x = 4

, so

x - 1 = 3

.

Tahmini Süre:1m 30s

Soru 228Soru →

For all $x > 0$ , which of the following is equivalent to the expression $\frac{4x^2 - 25}{2x^{\frac{3}{2}} + 5x^{\frac{1}{2}}}$ ?

2x - 5

2x^{\frac{3}{2}} - 5x^{\frac{1}{2}}

2x^{\frac{1}{2}} - 5x^{-\frac{1}{2}}

2x^{\frac{1}{2}} + 5x^{-\frac{1}{2}}

Cevabı ve açıklamayı göster

Cevap: $2x^{\frac{1}{2}} - 5x^{-\frac{1}{2}}$

Cevap

The correct equivalent expression is

2x^{\frac{1}{2}} - 5x^{-\frac{1}{2}}

.

To simplify the expression

\frac{4x^2 - 25}{2x^{\frac{3}{2}} + 5x^{\frac{1}{2}}}

, we first factor the numerator and the denominator. The numerator is a difference of squares:

4x^2 - 25 = (2x - 5)(2x + 5)

. In the denominator, we can factor out

x^{\frac{1}{2}}

to get

x^{\frac{1}{2}}(2x + 5)

. Substituting these factored forms gives

\frac{(2x - 5)(2x + 5)}{x^{\frac{1}{2}}(2x + 5)}

. Canceling the common factor

(2x + 5)

yields

\frac{2x - 5}{x^{\frac{1}{2}}}

. Dividing each term in the numerator by the denominator gives

\frac{2x}{x^{\frac{1}{2}}} - \frac{5}{x^{\frac{1}{2}}}

. Applying exponent rules, this simplifies to

2x^{1 - \frac{1}{2}} - 5x^{-\frac{1}{2}} = 2x^{\frac{1}{2}} - 5x^{-\frac{1}{2}}

, which is the correct expression.

Adım Adım Çözüm

1

Factor the numerator of the expression as a difference of squares.

4x^2 - 25 = (2x - 5)(2x + 5)

To simplify the rational expression, we need to factor both the numerator and the denominator to identify common factors.

2

Factor out the common term

x^{\frac{1}{2}}

from the denominator.

2x^{\frac{3}{2}} + 5x^{\frac{1}{2}} = x^{\frac{1}{2}}(2x + 5)

Factoring out

x^{\frac{1}{2}}

reveals the common binomial factor

(2x + 5)

in the denominator.

3

Substitute the factored forms back into the original expression and cancel the common factor

(2x + 5)

.

\frac{(2x - 5)(2x + 5)}{x^{\frac{1}{2}}(2x + 5)} = \frac{2x - 5}{x^{\frac{1}{2}}}

Since

x > 0

, the term

2x + 5

is non-zero, allowing us to cancel it from both the numerator and the denominator.

4

Divide each term in the numerator by

x^{\frac{1}{2}}

and apply the rules of exponents.

\frac{2x}{x^{\frac{1}{2}}} - \frac{5}{x^{\frac{1}{2}}} = 2x^{1 - \frac{1}{2}} - 5x^{-\frac{1}{2}} = 2x^{\frac{1}{2}} - 5x^{-\frac{1}{2}}

This simplifies the rational expression to its final equivalent form.

Anahtar Kavram

Equivalent Algebraic Expressions

Soru 229Soru →

If $\frac{8^x}{2^y} = 64$ , which of the following equations correctly expresses $y$ in terms of $x$ ?

y = 3x - 64

y = 3x - 6

y = 3x + 6

y = 6 - 3x

Cevabı ve açıklamayı göster

Cevap: $y = 3x - 6$

Cevap

y = 3x - 6

To solve the equation, express all terms using the common base of 2. Substituting

8 = 2^3

and

64 = 2^6

into the equation gives

\frac{(2^3)^x}{2^y} = 2^6

. Applying the power rule of exponents transforms the numerator into

2^{3x}

, which leads to the equation

\frac{2^{3x}}{2^y} = 2^6

. Using the quotient rule of exponents, the division simplifies to

2^{3x - y} = 2^6

. Since the bases are now identical, their exponents must be equal:

3x - y = 6

. Isolating the variable yields the correct formulation.

Adım Adım Çözüm

1

Rewrite all bases in the equation as powers of the common base 2.

Since

8 = 2^3

and

64 = 2^6

, the equation becomes

\frac{(2^3)^x}{2^y} = 2^6

.

Expressing all exponential terms using a single base allows for the application of exponent rules.

2

Apply the power of a power property to simplify the numerator on the left side.

The term

(2^3)^x

simplifies to

2^{3x}

, yielding the equation

\frac{2^{3x}}{2^y} = 2^6

.

The exponent rule

(a^m)^n = a^{mn}

states that a base raised to a power and then another power requires multiplying the exponents.

3

Apply the quotient property of exponents to combine the left side of the equation.

The left side simplifies to

2^{3x - y}

, resulting in

2^{3x - y} = 2^6

.

The quotient rule

\frac{a^m}{a^n} = a^{m-n}

states that dividing powers with the same base requires subtracting the exponent of the denominator from the exponent of the numerator.

4

Set the exponents equal to each other and solve for the desired variable.

3x - y = 6

, which simplifies to

y = 3x - 6

when solved for

y

.

Since the bases on both sides of the equation are equal, their exponents must also be equal.

Anahtar Kavram

Solving exponential equations by base conversion and rules of exponents

Alternatif Yöntem

Instead of converting both sides to base 2, you can also substitute a test value for

x

. For example, if

x = 2

, then the equation becomes

\frac{8^2}{2^y} = 64

, which simplifies to

\frac{64}{2^y} = 64 \implies 2^y = 1 \implies y = 0

. Substituting

x = 2

into the correct option yields

y = 3(2) - 6 = 0

, confirming the answer.

Tahmini Süre:1m 30s

Soru 230Soru →

The concentration of a medication in a patient's bloodstream, in milligrams per liter ( $\text{mg/L}$ ), $h$ hours after injection is modeled by an exponential decay function. If the concentration of the medication decreases by $48.8\%$ every $3$ hours, which of the following is closest to the percent decrease of the medication concentration each hour?

16.3%

20.0%

80.0%

86.6%

Cevabı ve açıklamayı göster

Cevap: 20.0%

Cevap

20.0%

The correct answer is 20.0%. Since the concentration decreases by 48.8% every 3 hours, the remaining concentration after 3 hours is 51.2%, which corresponds to a factor of 0.512 of the initial concentration. The hourly decay factor

b

satisfies the equation

b^3 = 0.512

. Taking the cube root of both sides yields

b = 0.8

. A decay factor of 0.8 means that 80% of the medication remains each hour, representing a percent decrease of 20% each hour.

Adım Adım Çözüm

1

Determine the remaining fraction of the medication after 3 hours.

Remaining fraction = 1 - 0.488 = 0.512

Since the concentration decreases by 48.8% every 3 hours, the remaining concentration is 51.2% of its value at the beginning of the 3-hour interval.

2

Set up the equation for the hourly retention factor,

b

.

b^3 = 0.512

An exponential decay model of the form

C(h) = C_0 \cdot b^h

retains a factor of

b^3

over a 3-hour period.

3

Solve for the hourly retention factor,

b

.

b = \sqrt[3]{0.512} = 0.8

Taking the cube root of 0.512 gives the hourly factor, indicating that the concentration retains 80% of its value each hour.

4

Calculate the hourly percent decrease from the hourly retention factor.

Percent decrease =

(1 - 0.8) \times 100\% = 20.0\%

The hourly percent decrease is the difference between the initial 100% and the remaining 80% retained each hour.

Anahtar Kavram

Converting decay factors across different time intervals using exponent properties.

Tahmini Süre:2m 0s

Soru 231Soru →

A quadratic function $f$ is defined by $f(x) = a(x - h)^2 + k$ , where $a$ , $h$ , and $k$ are constants, and its graph in the $xy$ -plane has vertex $(h, k)$ in the first quadrant. The $y$ -intercept of the graph of $f$ is $(0, 4)$ . The function $g$ is defined by $g(x) = f(x - 2) + 12$ . If the $y$ -intercept of the graph of $g$ is $(0, 8)$ , and the vertex of the graph of $f$ lies on the line $y = 5x$ , what is the value of $k$ ?

4

5

\frac{20}{3}

20

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

k

is 5.

By using the given vertex relation

k = 5h

, we can substitute this into the

y

-intercept equations for

f(x)

and

g(x)

. The

y

-intercept of

f

is at

(0,4)

, which translates to the equation

ah^2 + k = 4

. The

y

-intercept of

g

is at

(0,8)

, which leads to

a(h+2)^2 + k = -4

. Eliminating the parameter

a

by equating the two representations of

a

in terms of

h

yields the quadratic equation

3h^2 + h - 4 = 0

. Factoring this equation gives

(3h+4)(h-1)=0

. Since the vertex is in the first quadrant, both coordinates must be positive, which restricts

h

to the positive value

h = 1

. Substituting

h = 1

back into the vertex relation

k = 5h

yields

k = 5

.

Adım Adım Çözüm

1

Use the vertex line relation to express

k

in terms of

h

.

k = 5h

Since the vertex

(h, k)

lies on the line

y = 5x

, substituting the coordinates into the line's equation gives

k = 5h

.

2

Set up an equation using the

y

-intercept of

f(x)

.

ah^2 + 5h = 4 \implies a = \frac{4 - 5h}{h^2}

The

y

-intercept of

f(x)

is

(0, 4)

, so

f(0) = a(0-h)^2 + k = 4

. Substituting

k = 5h

gives

ah^2 + 5h = 4

.

3

Set up an equation using the

y

-intercept of

g(x)

.

a(h + 2)^2 + 5h = -4 \implies a = \frac{-5h - 4}{(h+2)^2}

The function

g(x) = f(x - 2) + 12 = a(x - 2 - h)^2 + k + 12

. The

y

-intercept is

(0, 8)

, so

g(0) = a(-2 - h)^2 + k + 12 = 8

. Simplifying and substituting

k = 5h

gives

a(h+2)^2 + 5h = -4

.

4

Equate the expressions for

a

and solve the resulting quadratic equation for

h

.

3h^2 + h - 4 = 0 \implies (3h + 4)(h - 1) = 0

Equating the two expressions for

a

gives

\frac{4 - 5h}{h^2} = \frac{-(5h + 4)}{(h + 2)^2}

. Cross-multiplying yields

(4 - 5h)(h^2 + 4h + 4) = -h^2(5h + 4) \implies 16 - 4h - 16h^2 - 5h^3 = -5h^3 - 4h^2 \implies 12h^2 + 4h - 16 = 0 \implies 3h^2 + h - 4 = 0

.

5

Determine

h

and compute

k

based on the quadrant constraints.

h = 1 \implies k = 5

Since the vertex

(h, k)

is in the first quadrant, we must have

h > 0

. Thus,

h = 1

is the only valid solution. Using

k = 5h

, we find

k = 5(1) = 5

.

Anahtar Kavram

Analyzing quadratic vertex form and transformations using algebraic systems.

Tahmini Süre:3m 0s

Soru 232Soru →

Consider the equation:

\frac{x + 3}{x - 1} + \frac{4}{x} = \frac{4}{x^2 - x}

What is the value of $x + 3$ for the real value of $x$ that satisfies this equation?

-5

4

11

2

Cevabı ve açıklamayı göster

Cevap: -5

Cevap

-5

The correct answer is

-5

. Multiplying the rational equation by the common denominator

x(x - 1)

gives the simplified equation

x(x + 3) + 4(x - 1) = 4

. Expanding and rearranging the terms leads to the quadratic equation

x^2 + 7x - 8 = 0

. Factoring this quadratic yields

(x + 8)(x - 1) = 0

, giving the potential solutions

x = -8

and

x = 1

. Substituting these values back into the original equation shows that

x = 1

causes division by zero, making it extraneous. Thus, the only valid solution is

x = -8

. Substituting this valid solution into the expression

x + 3

yields

-8 + 3 = -5

.

Adım Adım Çözüm

1

Find the common denominator of the rational terms.

The denominators are

x - 1

,

x

, and

x^2 - x

. The common denominator is

x(x - 1) = x^2 - x

.

Finding a common denominator allows us to eliminate the fractions by multiplying both sides.

2

Multiply the entire equation by the common denominator

x(x - 1)

to clear the denominators.

x(x + 3) + 4(x - 1) = 4

This simplifies the rational equation into a polynomial equation.

3

Expand and simplify the resulting equation.

x^2 + 3x + 4x - 4 = 4

, which simplifies to

x^2 + 7x - 4 = 4

.

Expanding the terms allows us to collect like terms and identify the type of polynomial equation.

4

Write the quadratic equation in standard form and factor it.

x^2 + 7x - 8 = 0

, which factors as

(x + 8)(x - 1) = 0

.

Setting the quadratic equation to zero allows us to find its roots by factoring.

5

Find the potential solutions and check for extraneous values in the original equation.

The potential solutions are

x = -8

and

x = 1

. The value

x = 1

is extraneous because it makes the denominators

x - 1

and

x^2 - x

equal to zero. The only valid solution is

x = -8

.

We must check if the roots make any denominator in the original equation zero, as division by zero is undefined.

6

Evaluate the expression

x + 3

using the valid solution.

-8 + 3 = -5

The question asks for the value of the expression

x + 3

, not the value of

x

itself.

Anahtar Kavram

Solving rational equations and checking for extraneous solutions

Soru 233Soru →

A parabola in the $xy$ -plane has equation $y = a(x - h)^2 + k$ , where $a$ , $h$ , and $k$ are constants. The parabola passes through the points $(2, 13)$ and $(8, 13)$ in the $xy$ -plane. If the minimum value of the quadratic function defined by this equation is $-5$ , what is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The correct answer is 2.

The correct answer is 2. The axis of symmetry of the parabola is halfway between the x-coordinates of the two symmetric points

(2, 13)

and

(8, 13)

, which is

x = 5

. Since the minimum value of the function is

-5

, the vertex of the parabola is

(5, -5)

. Writing the equation in vertex form,

y = a(x - 5)^2 - 5

, and substituting the point

(2, 13)

yields

13 = a(2 - 5)^2 - 5

, which simplifies to

18 = 9a

, so

a = 2

.

Adım Adım Çözüm

1

Find the axis of symmetry

x = h

using the symmetric points

(2, 13)

and

(8, 13)

.

h = 5

Since the points

(2, 13)

and

(8, 13)

have the same

y

-coordinate, the axis of symmetry of the parabola must lie exactly halfway between their

x

-coordinates:

h = \frac{2 + 8}{2} = 5

.

2

Determine the vertex

(h, k)

of the parabola.

(5, -5)

The minimum value of a quadratic function occurs at its vertex. The given minimum value is

-5

, which represents the

y

-coordinate (

k

) of the vertex. Combining this with

h = 5

, the vertex is

(5, -5)

.

3

Write the equation of the parabola in vertex form.

y = a(x - 5)^2 - 5

The vertex form of a quadratic equation is

y = a(x - h)^2 + k

. Substituting the values

h = 5

and

k = -5

yields this equation.

4

Substitute one of the given points into the equation to solve for

a

.

a = 2

Substituting

(2, 13)

into the equation gives

13 = a(2 - 5)^2 - 5 \Rightarrow 18 = 9a \Rightarrow a = 2

.

Anahtar Kavram

Finding the equation of a parabola using symmetry and vertex form

Tahmini Süre:2m 0s

Soru 234Soru →

What is the sum of all real solutions to the equation $\sqrt{3x + 7} - \frac{10}{\sqrt{3x + 7}} = 3$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The only real solution is 6, so the sum of all real solutions is 6.

The correct answer is 6. By substituting

u = \sqrt{3x + 7}

, the equation becomes

u - 10/u = 3

, which simplifies to the quadratic

u^2 - 3u - 10 = 0

. Factoring gives

(u - 5)(u + 2) = 0

. Since the principal square root must be non-negative,

u

must be positive, so we reject

u = -2

and keep

u = 5

. Solving

\sqrt{3x + 7} = 5

by squaring both sides yields

3x + 7 = 25

, which simplifies to

x = 6

. The extraneous solution

x = -1

(which comes from

u = -2

) must be discarded because substituting it back into the original equation results in

-3

instead of

3

. Thus, the only real solution is 6.

Adım Adım Çözüm

1

Substitute

u = \sqrt{3x + 7}

into the equation.

u - \frac{10}{u} = 3

, where

u > 0

.

This substitution simplifies the radical equation into a rational equation of a single variable.

2

Multiply the entire equation by

u

and solve the resulting quadratic equation.

u^2 - 3u - 10 = 0 \implies (u - 5)(u + 2) = 0

, which yields

u = 5

or

u = -2

.

Clearing the denominator transforms the rational expression into a standard quadratic equation.

3

Apply the constraint that

u = \sqrt{3x + 7}

must be non-negative.

u = 5

is the only valid root;

u = -2

has no real solution and must be discarded.

The output of a principal square root is non-negative by definition.

4

Substitute back

\sqrt{3x + 7} = 5

and solve for

x

.

3x + 7 = 25 \implies 3x = 18 \implies x = 6

.

Squaring both sides eliminates the radical and allows us to isolate the variable

x

.

5

Verify if solving

\sqrt{3x+7} = -2

leads to an extraneous solution.

Squaring both sides of

\sqrt{3x+7} = -2

gives

3x+7 = 4 \implies x = -1

. Substituting

x = -1

into the original equation yields

\sqrt{4} - \frac{10}{\sqrt{4}} = 2 - 5 = -3 \neq 3

. Therefore,

x = -1

is extraneous.

Verifying solutions is necessary to identify and discard values that do not satisfy the original equation.

Anahtar Kavram

Solving equations using substitution and identifying extraneous solutions

Tahmini Süre:2m 30s

Soru 235Soru →

If $x$ is a real solution to the equation $\sqrt{2x + 7} - x = 2$ , what is the value of $x + 4$ ?

5

1

7

3

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The correct answer is 5. To solve the equation

\sqrt{2x + 7} - x = 2

, we first isolate the radical to get

\sqrt{2x + 7} = x + 2

. Squaring both sides yields

2x + 7 = x^2 + 4x + 4

. Rearranging this equation into standard quadratic form gives

x^2 + 2x - 3 = 0

, which factors as

(x + 3)(x - 1) = 0

. This gives two potential solutions:

x = 1

and

x = -3

. Checking these in the original equation shows that

x = 1

is valid, whereas

x = -3

is extraneous because

\sqrt{2(-3)+7} - (-3) = 1 + 3 = 4 \neq 2

. Therefore, the only real solution is

x = 1

. Substituting this into the expression

x + 4

gives

1 + 4 = 5

.

Adım Adım Çözüm

1

Isolate the radical expression on one side of the equation.

\sqrt{2x + 7} = x + 2

This sets up the equation to eliminate the radical by squaring both sides.

2

Square both sides of the equation to eliminate the radical.

2x + 7 = (x + 2)^2

Squaring a square root removes the radical, allowing the equation to be solved algebraically.

3

Expand the squared binomial on the right side.

2x + 7 = x^2 + 4x + 4

Expanding

(x+2)^2

yields a quadratic term, a linear term, and a constant term.

4

Rearrange the equation to set it equal to zero.

x^2 + 2x - 3 = 0

Subtracting

2x

and

7

from both sides simplifies the equation into standard quadratic form.

5

Factor the quadratic equation.

(x + 3)(x - 1) = 0

Factoring allows us to find the potential solutions for

x

.

6

Identify potential solutions and check for extraneous solutions by substituting them back into the original equation.

For

x = 1

,

\sqrt{2(1) + 7} - 1 = 3 - 1 = 2

, which is true. For

x = -3

,

\sqrt{2(-3) + 7} - (-3) = 1 + 3 = 4

, which does not equal

2

.

Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original radical equation. Thus,

x = -3

is extraneous, leaving

x = 1

as the only valid solution.

7

Calculate the value of the expression

x + 4

using the valid solution.

1 + 4 = 5

The question asks for the value of

x + 4

, so we substitute the valid solution

x = 1

into this expression.

Anahtar Kavram

Solving radical equations and checking for extraneous solutions.

Soru 236Soru →

The graph of the polynomial function $f$ in the $xy$ -plane is defined by $f(x) = a(x - 3)(x + 4)$ , where $a$ is a constant. If the $y$ -intercept of the graph is $(0, -24)$ , what is the value of $a$ ?

-2

\frac{1}{2}

2

-12

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2

The correct answer is 2. The

y

-intercept of a graph is the point where

x = 0

. For the function

f(x) = a(x - 3)(x + 4)

, substituting

x = 0

yields

f(0) = a(0 - 3)(0 + 4) = a(-3)(4) = -12a

. Given that the

y

-intercept is

(0, -24)

, the value of the function at

x = 0

is

-24

. Setting these two values equal gives the equation

-12a = -24

. Dividing both sides of the equation by

-12

isolates the constant, resulting in

a = 2

.

Adım Adım Çözüm

1

Identify the relationship between the

y

-intercept and the function value at

x = 0

.

Since the

y

-intercept is

(0, -24)

, it follows that

f(0) = -24

.

By definition, the

y

-intercept of a graph is the point where the graph crosses the vertical axis, which occurs at

x = 0

.

2

Substitute

x = 0

into the polynomial expression for

f(x)

.

f(0) = a(0 - 3)(0 + 4) = a(-3)(4) = -12a

.

This evaluates the algebraic expression at the

y

-intercept to express the value in terms of the constant

a

.

3

Equate the evaluated expression to the known

y

-intercept value and solve for

a

.

-12a = -24 \Rightarrow a = 2

.

Setting the two expressions for

f(0)

equal allows us to solve the linear equation for the constant

a

.

Anahtar Kavram

Evaluating a polynomial at

x = 0

to relate its algebraic form to its

y

-intercept.

Soru 237Soru →

What is the only real solution to the equation $\sqrt{3x + 13} = x + 3$ ?

-4

Both

-4

and

1

1

-1

Cevabı ve açıklamayı göster

Cevap: $1$

Cevap

The correct answer is the value 1.

Squaring both sides of the equation

\sqrt{3x + 13} = x + 3

yields

3x + 13 = x^2 + 6x + 9

. Rearranging this into standard quadratic form gives

x^2 + 3x - 4 = 0

. Factoring the quadratic expression gives

(x + 4)(x - 1) = 0

, which yields the potential solutions

x = 1

and

x = -4

. Substituting

x = 1

back into the original equation gives

\sqrt{16} = 4

, which is true. Therefore, the value

1

is the only real solution.

Adım Adım Çözüm

1

Square both sides of the equation to eliminate the radical.

3x + 13 = (x + 3)^2

Squaring both sides eliminates the radical, allowing us to solve for

x

.

2

Expand the right side and rewrite the equation in standard quadratic form.

x^2 + 3x - 4 = 0

Expanding

(x+3)^2

yields

x^2 + 6x + 9

. Subtracting

3x

and

13

from both sides sets the quadratic equation to zero.

3

Factor the quadratic equation.

(x + 4)(x - 1) = 0

We find two numbers that multiply to

-4

and add to

3

, which are

4

and

-1

.

4

Solve for the potential solutions.

x = -4

or

x = 1

Setting each factor to zero yields the potential solutions.

5

Verify each potential solution in the original equation.

x = 1

is valid, and

x = -4

is extraneous.

Plugging in

x = 1

gives

\sqrt{16} = 4

, which is true. Plugging in

x = -4

gives

\sqrt{1} = -1

, which is false since the principal square root is non-negative.

Anahtar Kavram

Solving radical equations and checking for extraneous solutions.

Tahmini Süre:45s

Soru 238Soru →

If $x$ and $y$ satisfy the system of equations below, what is the value of $x + y$ ?

\begin{aligned} 8^{x-y} &= 2^{y+3} \\ 9^{x} &= 27^{y-1} \end{aligned}

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

36

The correct answer is 36. By rewriting

8^{x-y} = 2^{y+3}

with base 2, we obtain the relation

3x - 4y = 3

. Similarly, rewriting

9^x = 27^{y-1}

with base 3 gives

2x - 3y = -3

. Solving this system of equations simultaneously yields

x = 21

and

y = 15

. Adding these values together gives the final result of 36.

Adım Adım Çözüm

1

Rewrite the first equation using the common base of 2.

2^{3(x-y)} = 2^{y+3}

, which simplifies to

3x - 4y = 3

.

Since

8 = 2^3

, we can apply the power of a power rule

(a^m)^n = a^{mn}

to rewrite

8^{x-y}

as

2^{3(x-y)}

. Setting the exponents equal gives the linear equation

3(x-y) = y+3

, which simplifies to

3x - 4y = 3

.

2

Rewrite the second equation using the common base of 3.

3^{2x} = 3^{3(y-1)}

, which simplifies to

2x - 3y = -3

.

Since

9 = 3^2

and

27 = 3^3

, we rewrite the terms using base 3. Equating the exponents gives

2x = 3(y-1)

, which simplifies to

2x - 3y = -3

.

3

Solve the system of two linear equations.

x = 21

and

y = 15

.

Using the elimination method, we can multiply the first equation by 3 and the second equation by 4 to align the coefficients of

y

, obtaining

9x - 12y = 9

and

8x - 12y = -12

. Subtracting the equations gives

x = 21

. Substituting this back yields

y = 15

.

4

Find the sum of

x

and

y

.

36

The question asks for the value of

x + y

. Substituting

x = 21

and

y = 15

gives

21 + 15 = 36

.

Anahtar Kavram

Solving systems of exponential equations by expressing bases as powers of common bases to establish a system of linear equations.

Soru 239Soru →

If $\sqrt{4x + 9} = 5$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

Squaring both sides of the equation

\sqrt{4x + 9} = 5

yields

4x + 9 = 25

. Subtracting 9 from both sides gives

4x = 16

, and dividing by 4 yields

x = 4

. Checking this solution in the original equation yields

\sqrt{4(4) + 9} = \sqrt{25} = 5

, which confirms it is correct.

Adım Adım Çözüm

1

Square both sides of the equation

4x + 9 = 25

To eliminate the square root on the left side of the equation.

2

Subtract 9 from both sides of the equation

4x = 16

To isolate the variable term on the left side of the equation.

3

Divide both sides of the equation by 4

x = 4

To solve for

x

.

4

Substitute the solved value back into the original equation

\sqrt{4(4) + 9} = 5 \implies 5 = 5

To ensure the solution is valid and not extraneous.

Anahtar Kavram

Solving radical equations of the form

\sqrt{ax + b} = c

by squaring both sides.

Soru 240Soru →

The polynomial function $p$ is defined by $p(x) = (x - 3)(x - 5)(x - r)$ , where $r$ is a constant. If the graph of $y = p(x)$ in the $xy$ -plane intersects the $y$ -axis at $(0, -60)$ , what is the value of $r$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

r

is

4

.

Evaluating the polynomial at

x = 0

yields the

y

-intercept of the graph, which is given as

-60

. Substituting

0

for

x

in the equation

p(x) = (x - 3)(x - 5)(x - r)

gives

p(0) = (0 - 3)(0 - 5)(0 - r) = (-3)(-5)(-r) = -15r

. Setting this expression equal to

-60

yields

-15r = -60

. Dividing both sides of the equation by

-15

yields

r = 4

.

Adım Adım Çözüm

1

Identify the meaning of the

y

-intercept.

Evaluating

p(x)

at

x = 0

must equal

-60

, so

p(0) = -60

.

The

y

-intercept of a graph occurs where the

x

-coordinate is

0

.

2

Substitute

x = 0

into the polynomial expression.

p(0) = (0 - 3)(0 - 5)(0 - r) = (-3)(-5)(-r) = -15r

.

This simplifies the polynomial expression in terms of the unknown constant

r

.

3

Set the expression equal to the given

y

-intercept value and solve for

r

.

-15r = -60 \implies r = 4

.

Dividing both sides by

-15

isolates the variable

r

.

Anahtar Kavram

Using the y-intercept of a polynomial function to find an unknown root/constant in its factored form.

Advanced Math

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar