Tüm alıştırma soruları

Soru 2021Soru →

For all non-zero real numbers

a

,

b

, and

c

, the expression below is simplified:

\frac{(2a^2 b^{-1} c^3)^3}{4a^5 (b^2 c^{-2})^{-2}}

Which of the following is equivalent to this expression?

\frac{2ac^5}{b^7}

\frac{abc^5}{2}

2abc^5

4abc^5

2b^2c^{10}

Cevabı ve açıklamayı göster

Cevap: $2abc^5$

Cevap

The expression is equivalent to

2abc^5

.

The correct answer is obtained by first simplifying the numerator and denominator using the power of a product and power of a power rules, then dividing the coefficients and subtracting the exponents of like bases. This yields the simplified expression

2abc^5

.

Adım Adım Çözüm

1

Apply the power of a product rule to the numerator.

(2a^2 b^{-1} c^3)^3 = 2^3 \cdot (a^2)^3 \cdot (b^{-1})^3 \cdot (c^3)^3 = 8 a^6 b^{-3} c^9

Each factor inside the parentheses must be raised to the power of 3, multiplying the exponents of the variables.

2

Apply the power of a product rule to the denominator's parentheses.

4a^5 (b^2 c^{-2})^{-2} = 4a^5 \cdot (b^2)^{-2} \cdot (c^{-2})^{-2} = 4a^5 b^{-4} c^4

The terms inside the parentheses are raised to the power of -2, multiplying their exponents.

3

Divide the simplified numerator by the simplified denominator.

2a^1 b^1 c^5

, which is

2abc^5

Divide the coefficients (8 / 4 = 2) and subtract the exponents of the same bases:

a^{6-5} = a^1

,

b^{-3 - (-4)} = b^1

, and

c^{9-4} = c^5

.

Anahtar Kavram

Properties of Exponents in Algebraic Expressions

Alternatif Yöntem

Alternatively, you can rewrite the expression by eliminating negative exponents first. Convert

b^{-1}

to

\frac{1}{b}

,

b^2

to itself, and

c^{-2}

to

\frac{1}{c^2}

within the parentheses, apply the outer exponents, and then simplify the resulting complex fraction.

Tahmini Süre:1m 30s

Soru 2022Soru →

Two polynomial expressions are defined as $P(x) = (2x - 4)(x^2 - 5x + c)$ and $Q(x) = (x^2 + 3x - 4)(2x + a)$ , where $a$ and $c$ are constants. When $P(x)$ is expanded and simplified, it has no $x$ term. If the coefficient of the $x^2$ term in the expanded and simplified form of $Q(x)$ is equal to the coefficient of the $x^2$ term in the expanded and simplified form of $P(x)$ , what is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: -20

Cevap

The value of

a

is

-20

.

Expanding

P(x) = (2x - 4)(x^2 - 5x + c)

yields

2x^3 - 14x^2 + (2c + 20)x - 4c

. Since there is no

x

term,

2c + 20 = 0

, which means

c = -10

. This leaves the coefficient of the

x^2

term in

P(x)

as

-14

. Expanding

Q(x) = (x^2 + 3x - 4)(2x + a)

yields

2x^3 + (a + 6)x^2 + (3a - 8)x - 4a

, so the coefficient of its

x^2

term is

a + 6

. Setting

a + 6 = -14

and solving for

a

gives

a = -20

.

Adım Adım Çözüm

1

Expand the polynomial

P(x) = (2x - 4)(x^2 - 5x + c)

using the distributive property.

P(x) = 2x^3 - 14x^2 + (2c + 20)x - 4c

To identify the coefficients of each term in

P(x)

.

2

Set the coefficient of the

x

term,

2c + 20

, to

0

and solve for

c

.

c = -10

The problem states that

P(x)

has no

x

term when simplified, which means its coefficient must be zero.

3

Determine the coefficient of the

x^2

term in

P(x)

.

The coefficient of

x^2

is

-14

.

This coefficient will be equated to the

x^2

coefficient of

Q(x)

as per the problem constraints.

4

Expand the polynomial

Q(x) = (x^2 + 3x - 4)(2x + a)

using the distributive property.

Q(x) = 2x^3 + (a + 6)x^2 + (3a - 8)x - 4a

To identify the coefficient of the

x^2

term in

Q(x)

.

5

Set the coefficient of the

x^2

term in

Q(x)

, which is

a + 6

, equal to the coefficient of the

x^2

term in

P(x)

, which is

-14

, and solve for

a

.

a = -20

The problem states that these two coefficients are equal.

Anahtar Kavram

Operations on Polynomials

Tahmini Süre:2m 30s

Soru 2023Soru →

In the standard $(x, y)$ coordinate plane, a circle is defined by the equation $x^2 + y^2 = 17$ and a line is defined by the equation $y = x - 3$ . If $(x, y)$ represents the intersection point of the circle and the line that lies in the first quadrant, what is the value of $x + y$ ?

1

3

5

7

-5

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The sum of the coordinates of the first-quadrant intersection point is 5.

The correct answer is the sum of the coordinates of the first-quadrant intersection point. By substituting

y = x - 3

into the circle's equation, we get

x^2 + (x - 3)^2 = 17

. Expanding the squared term gives

x^2 + x^2 - 6x + 9 = 17

, which simplifies to

2x^2 - 6x - 8 = 0

. Dividing the entire equation by 2 yields

x^2 - 3x - 4 = 0

. Factoring this quadratic equation gives

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

, which has solutions

x = 4

and

x = -1

. Because the intersection point must lie in the first quadrant, the

x

-coordinate must be positive, so we choose

x = 4

. Substituting

x = 4

back into the linear equation gives

y = 4 - 3 = 1

. The sum of these coordinates is

4 + 1 = 5

.

Adım Adım Çözüm

1

Substitute the expression for

y

from the linear equation into the circle's equation.

x^2 + (x - 3)^2 = 17

This eliminates

y

so that we can solve for

x

.

2

Expand the squared binomial and combine like terms to write the equation in standard quadratic form.

2x^2 - 6x - 8 = 0

Expanding

(x - 3)^2

gives

x^2 - 6x + 9

, and adding

x^2

and subtracting 17 from both sides gives the quadratic equation.

3

Simplify the quadratic equation by dividing all terms by 2, then factor the resulting quadratic expression.

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

Dividing by 2 gives

x^2 - 3x - 4 = 0

, which factors into

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

since

(-4) \times 1 = -4

and

-4 + 1 = -3

.

4

Solve for

x

and select the positive solution since the point lies in the first quadrant.

x = 4

The solutions are

x = 4

and

x = -1

. In the first quadrant, both coordinates must be positive, so we choose

x = 4

.

5

Substitute

x = 4

back into the linear equation to find the corresponding

y

-coordinate.

y = 1

Using

y = x - 3

with

x = 4

gives

y = 4 - 3 = 1

.

6

Calculate the sum of the coordinates

x + y

.

4 + 1 = 5

The question asks for the value of

x + y

.

Anahtar Kavram

Solving a system consisting of a linear equation and a quadratic circle equation by substitution, factoring the resulting quadratic equation, and applying quadrant constraints.

Soru 2024Soru →

A rectangle has a length of $y + 7$ inches and a width of $y - 3$ inches. Which of the following expressions represents the area, in square inches, of the rectangle?

y^2 - 21

y^2 + 4y + 21

y^2 + 4y - 21

y^2 - 4y - 21

5y - 21

Cevabı ve açıklamayı göster

Cevap: y^2 + 4y - 21

Cevap

The expression

y^2 + 4y - 21

The area of a rectangle is found by multiplying its length by its width. The product of the dimensions is represented by the expression

(y + 7)(y - 3)

. Applying the distributive property gives

y^2 - 3y + 7y - 21

. Combining the like terms

-3y

and

7y

results in

+4y

. Therefore, the simplified expression for the area is

y^2 + 4y - 21

.

Adım Adım Çözüm

1

Set up the area formula for the rectangle by multiplying its length and width.

Area

= (y + 7)(y - 3)

The area of a rectangle is calculated by multiplying its length by its width.

2

Use the distributive property to expand the product of the two binomials.

Area

= y(y) + y(-3) + 7(y) + 7(-3) = y^2 - 3y + 7y - 21

Each term in the first binomial must be multiplied by each term in the second binomial.

3

Combine the like terms to simplify the expression into standard form.

Area

= y^2 + 4y - 21

Combining

-3y

and

+7y

yields

+4y

.

Anahtar Kavram

Finding the area of a rectangle by multiplying binomial expressions using the distributive property.

Tahmini Süre:45s

Soru 2025Soru →

Two functions, $f$ and $g$ , are defined as $f(x) = (x - 4)^2$ and $g(x) = |2x - 3|$ . What is the value of the composite function $f(g(x))$ evaluated at $x = -1$ ?

1

9

47

81

125

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

1

To find

f(g(-1))

, we first evaluate the inner function

g

at

x = -1

. Substituting

-1

into

g(x) = |2x - 3|

gives

g(-1) = |2(-1) - 3| = |-5| = 5

. Next, we use this result as the input for the outer function

f

, evaluating

f(5)

. Substituting

5

into

f(x) = (x - 4)^2

gives

f(5) = (5 - 4)^2 = 1^2 = 1

. This corresponds to the option with value 1.

Adım Adım Çözüm

1

Evaluate the inner function

g(-1)

first.

g(-1) = |2(-1) - 3| = |-2 - 3| = |-5| = 5

The input to the outer function of a composition is the output of the inner function.

2

Evaluate the outer function

f(x)

at the result from the previous step.

f(5) = (5 - 4)^2 = 1^2 = 1

We substitute the value of

g(-1)

, which is

5

, into the function

f(x)

to find the final value of the composition.

Anahtar Kavram

Evaluating a composite function involves applying the inner function first, then using that output as the input for the outer function.

Soru 2026Soru →

A construction company is working on two building projects, Project X and Project Y. The quantities of concrete (in tons) and steel (in tons) required for these projects are represented by the matrix $Q$ :

Q = \begin{bmatrix} 80 & 25 \\ 120 & 40 \end{bmatrix}

where the first row represents Project X, the second row represents Project Y, the first column represents concrete, and the second column represents steel.

The purchase cost per ton and the transportation cost per ton for these materials are represented by the matrix $C$ :

C = \begin{bmatrix} 110 & 15 \\ 650 & 45 \end{bmatrix}

where the first row represents concrete, the second row represents steel, the first column represents the purchase cost (in dollars per ton), and the second column represents the transportation cost (in dollars per ton).

If the product matrix $P = QC$ represents the total cost details for the two projects, what is the total transportation cost for Project Y, in dollars?

Cevabı ve açıklamayı göster

Cevap: 3600

Cevap

The total transportation cost for Project Y is 3,600 dollars.

The correct answer is 3,600. To find the total transportation cost for Project Y, we calculate the entry in the second row (Project Y) and second column (transportation cost) of the product matrix

P = QC

. This is calculated as

120 \text{ tons of concrete} \times 15 \text{ dollars per ton} + 40 \text{ tons of steel} \times 45 \text{ dollars per ton} = 1,800 + 1,800 = 3,600

dollars.

Adım Adım Çözüm

1

Identify the row representing Project Y and the column representing transportation cost

Row 2 of

Q

is

[120, 40]

and Column 2 of

C

is

[15, 45]^T

.

To find the total transportation cost for Project Y, we must compute the entry in the second row (Project Y) and second column (transportation cost) of the product matrix

P = QC

.

2

Multiply the row elements by the corresponding column elements

120 \times 15 = 1800

and

40 \times 45 = 1800

This calculates the individual transportation costs for the concrete and steel required for Project Y.

3

Sum the products to find the total transportation cost

1800 + 1800 = 3600

Adding these individual costs gives the total transportation cost for Project Y.

Anahtar Kavram

Matrix multiplication involves multiplying the elements of each row of the first matrix by the corresponding elements of each column of the second matrix and summing the products.

Soru 2027Soru →

If $3^{2x - 1} = 27$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

x

is

2

.

Rewriting

27

as

3^3

gives the equation

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

. Equating the exponents yields

2x - 1 = 3

, which solves to

x = 2

.

Adım Adım Çözüm

1

Rewrite the right side of the equation with a base of

3

.

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

Expressing both sides of the equation with a common base allows for direct comparison of the exponents.

2

Set the exponents equal to each other.

2x - 1 = 3

Since the bases are both

3

, their exponents must be equal for the expressions to be equal.

3

Solve the linear equation for

x

.

x = 2

Adding

1

to both sides gives

2x = 4

. Dividing both sides by

2

results in

x = 2

.

Anahtar Kavram

Solving exponential equations by expressing both sides with a common base

Soru 2028Soru →

If $x$ is a real number such that $\frac{(3^x)^4 \cdot 9^{2-x}}{27^{x-1}} = 81$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

x

is 3.

Rewriting the bases in terms of 3, the expression becomes

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

. Combining the numerator using the product rule gives

3^{2x+4}

in the numerator. Dividing by the denominator using the quotient rule gives

3^{(2x+4)-(3x-3)} = 3^{-x+7}

. Equating this to

3^4

gives

-x + 7 = 4

, which yields

x = 3

.

Adım Adım Çözüm

1

Express each base in terms of the common base 3

9 = 3^2

,

27 = 3^3

, and

81 = 3^4

Converting all terms to a common base allows the use of exponent rules to combine them.

2

Apply the power of a power rule

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

to rewrite each exponent

(3^x)^4 = 3^{4x}

,

(3^2)^{2-x} = 3^{4-2x}

, and

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

This simplifies individual terms by multiplying their exponents.

3

Apply the product rule of exponents

b^m \cdot b^n = b^{m+n}

to combine the terms in the numerator

3^{4x} \cdot 3^{4-2x} = 3^{4x + 4 - 2x} = 3^{2x + 4}

Multiplying exponential terms with the same base is simplified by adding their exponents.

4

Apply the quotient rule of exponents

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

to simplify the fraction

3^{(2x+4) - (3x-3)} = 3^{-x+7}

Dividing exponential terms with the same base is simplified by subtracting the exponent in the denominator from the exponent in the numerator.

5

Equate the exponents of the simplified base 3 expression and base 3 representation of 81

-x + 7 = 4

, which solves to

x = 3

Since the bases are equal, the powers must be equal for the equation to hold true.

Anahtar Kavram

Properties of Exponents in Algebraic Expressions

Soru 2029Soru →

An entrepreneur starts a company with an operating budget of $100,000$ in its first year. For each of the next 4 years (years 2 through 5), the budget increases by a constant amount of $d$ dollars each year. For years 6 through 8, the budget increases geometrically, where the budget in year 6 is $1.5$ times the budget in year 5, and the budget increases by $50\%$ each year thereafter. If the total operating budget over the first 8 years is $1,597,500$ dollars, what is the value of $d$ ?

4,300

10,000

17,500

33,900

38,500

Cevabı ve açıklamayı göster

Cevap: $10,000$

Cevap

The common difference is

10,000

.

The correct answer is found by setting up the sum of the first 5 years of the arithmetic sequence and the subsequent 3 years of the geometric sequence. Summing the expressions for all 8 years gives

1,212,500 + 38.5d

. Equating this expression to the total budget of

1,597,500

and solving for

d

yields

10,000

.

Adım Adım Çözüm

1

Express the budget for the first 5 years as an arithmetic sequence and find their sum.

S_5 = 100,000 + (100,000 + d) + (100,000 + 2d) + (100,000 + 3d) + (100,000 + 4d) = 500,000 + 10d

The budget starts at

100,000

in year 1 and increases by a constant amount

d

each year through year 5.

2

Express the budgets for years 6 through 8 as a geometric sequence starting from

1.5

times the year 5 budget.

Year 6:

1.5(100,000 + 4d) = 150,000 + 6d

; Year 7:

1.5(150,000 + 6d) = 225,000 + 9d

; Year 8:

1.5(225,000 + 9d) = 337,500 + 13.5d

. The sum of these 3 years is

712,500 + 28.5d

.

The budget increases by a factor of

1.5

(or

50\%

) each year starting from year 6.

3

Combine the sums of both sequences to represent the total 8-year budget and solve for

d

.

Total =

(500,000 + 10d) + (712,500 + 28.5d) = 1,212,500 + 38.5d

. Setting this equal to the given total:

1,212,500 + 38.5d = 1,597,500 \implies 38.5d = 385,000 \implies d = 10,000

.

The total budget over the 8 years is the sum of the budgets of the individual years.

Anahtar Kavram

Combining arithmetic and geometric sequences in multi-step word problems.

Alternatif Yöntem

Instead of calculating each geometric year sequentially, the sum of the geometric sequence for years 6 to 8 can be calculated using the geometric series sum formula

S_n = a_1 \frac{1 - r^n}{1 - r}

with

a_1 = 1.5(100,000 + 4d)

and

r = 1.5

over

n = 3

terms.

Tahmini Süre:2m 30s

Soru 2030Soru →

A local farm sells organic fruit baskets at a weekend market. The number of small and large baskets sold on Saturday and Sunday is represented by matrix $Q$ , where row 1 represents Saturday, row 2 represents Sunday, column 1 represents small baskets, and column 2 represents large baskets:

Q = \begin{bmatrix} 40 & 30 \\ 20 & 50 \end{bmatrix}

The selling price and the production cost, in dollars, for each type of basket are represented by matrix $P$ , where row 1 represents small baskets, row 2 represents large baskets, column 1 represents the selling price, and column 2 represents the production cost:

P = \begin{bmatrix} 15 & 8 \\ 25 & 12 \end{bmatrix}

The product matrix $R = QP$ represents the total revenue and total production cost for each day. Which of the following matrices represents $R$ ?

\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}

\begin{bmatrix} 600 & 240 \\ 500 & 600 \end{bmatrix}

\begin{bmatrix} 760 & 850 \\ 1240 & 1350 \end{bmatrix}

\begin{bmatrix} 1350 & 1550 \\ 680 & 760 \end{bmatrix}

\begin{bmatrix} 1450 & 680 \\ 1550 & 660 \end{bmatrix}

Cevabı ve açıklamayı göster

Cevap: $\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}$

Cevap

\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}

The correct product matrix is obtained by performing matrix multiplication

QP

. Multiplying the first row of

Q

by the first column of

P

yields the Saturday revenue:

40 \times 15 + 30 \times 25 = 1350

. Multiplying the first row of

Q

by the second column of

P

yields the Saturday cost:

40 \times 8 + 30 \times 12 = 680

. Repeating this for the second row of

Q

(Sunday) yields the Sunday revenue of

1550

and Sunday cost of

760

. Thus, the resulting matrix is the one with row 1 equal to

[1350, 680]

and row 2 equal to

[1550, 760]

.

Adım Adım Çözüm

1

Set up the matrix multiplication product

R = QP

.

R = \begin{bmatrix} 40 & 30 \\ 20 & 50 \end{bmatrix} \begin{bmatrix} 15 & 8 \\ 25 & 12 \end{bmatrix}

To find the total revenue and production cost for Saturday and Sunday, we must multiply the quantity matrix by the price-cost matrix.

2

Calculate the elements of the first row of the product matrix

R

, representing Saturday's revenue and cost.

Row 1, Column 1 (Saturday Revenue):

40(15) + 30(25) = 600 + 750 = 1350

. Row 1, Column 2 (Saturday Cost):

40(8) + 30(12) = 320 + 360 = 680

.

Multiply the first row of matrix

Q

by the columns of matrix

P

to determine Saturday's financial values.

3

Calculate the elements of the second row of the product matrix

R

, representing Sunday's revenue and cost.

Row 2, Column 1 (Sunday Revenue):

20(15) + 50(25) = 300 + 1250 = 1550

. Row 2, Column 2 (Sunday Cost):

20(8) + 50(12) = 160 + 600 = 760

.

Multiply the second row of matrix

Q

by the columns of matrix

P

to determine Sunday's financial values.

4

Combine the calculated elements into the final

2 \times 2

matrix.

\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}

Placing the values in their corresponding row and column positions yields the complete product matrix.

Anahtar Kavram

Matrix Multiplication in Applied Word Problems

Tahmini Süre:1m 30s

Soru 2031Soru →

A line intersects a parabola at two distinct points in the standard $(x, y)$ coordinate plane. The system of equations representing these curves is given by:

\begin{aligned} x^2 - 7x + 3y &= 6 \\ 3y - 4x &= 6 \end{aligned}

What is the distance between the two intersection points?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

Solving the system of equations yields the intersection points

(0, 2)

and

(3, 6)

. The distance between these points is computed using the distance formula:

\sqrt{(3 - 0)^2 + (6 - 2)^2} = \sqrt{25} = 5

.

Adım Adım Çözüm

1

Isolate the

3y

term in the linear equation.

3y = 4x + 6

This allows for a direct substitution into the quadratic equation without introducing fractional coefficients.

2

Substitute

3y

into the first equation and simplify.

x^2 - 3x = 0

Substituting

4x + 6

for

3y

yields

x^2 - 7x + (4x + 6) = 6

. Subtracting 6 from both sides and combining like terms simplifies this to a basic quadratic equation.

3

Solve the quadratic equation for

x

and determine the corresponding

y

-coordinates.

The intersection points are

(0, 2)

and

(3, 6)

.

Factoring gives

x(x - 3) = 0

, so the

x

-coordinates are

0

and

3

. Substituting

x = 0

into the linear relation gives

3y = 6 \implies y = 2

. Substituting

x = 3

gives

3y = 18 \implies y = 6

.

4

Calculate the distance between the two coordinates.

5

Using the distance formula:

d = \sqrt{(3 - 0)^2 + (6 - 2)^2} = \sqrt{9 + 16} = 5

.

Anahtar Kavram

Systems of Linear and Non-Linear Equations

Soru 2032Soru →

What is the greatest integer value of $x$ that satisfies the inequality $\frac{2 - 5x}{3} - \frac{x - 4}{2} \geq 3$ ?

Cevabı ve açıklamayı göster

Cevap: -1

Cevap

The greatest integer value of

x

that satisfies the inequality is -1.

The correct answer is -1 because solving the inequality leads to

x \leq -\frac{2}{13}

. Since

-\frac{2}{13}

is approximately

-0.154

, the set of integers satisfying the inequality is

\{-1, -2, -3, \dots\}

. The greatest integer in this set is -1.

Adım Adım Çözüm

1

Multiply the entire inequality by the least common multiple of the denominators (6) to eliminate the fractions.

2(2 - 5x) - 3(x - 4) \geq 18

Multiplying by a positive number clears the fractions without changing the direction of the inequality.

2

Distribute the coefficients and combine like terms on the left side of the inequality.

16 - 13x \geq 18

Simplifying the expressions on each side makes it easier to isolate the variable.

3

Subtract 16 from both sides to isolate the term with the variable

x

.

-13x \geq 2

Moving the constant terms to one side prepares the inequality for division.

4

Divide both sides by -13 and reverse the direction of the inequality sign.

x \leq -\frac{2}{13}

Dividing by a negative number requires flipping the inequality sign to maintain a true statement.

5

Determine the greatest integer that is less than or equal to

-\frac{2}{13}

.

-1

Since

-\frac{2}{13} \approx -0.154

, the largest integer that is less than or equal to this value is -1.

Anahtar Kavram

Solving multi-step linear inequalities, including clearing fractional coefficients and reversing the inequality sign when multiplying or dividing by a negative number.

Soru 2033Soru →

What is the value of the discriminant of the quadratic equation $3x^2 + 5x - 2 = 0$ ?

Cevabı ve açıklamayı göster

Cevap: 49

Cevap

The discriminant of the quadratic equation is 49.

The discriminant is calculated using the formula

b^2 - 4ac

. For the equation

3x^2 + 5x - 2 = 0

, the coefficients are

a = 3

,

b = 5

, and

c = -2

. Substituting these yields

5^2 - 4(3)(-2) = 25 - (-24) = 25 + 24 = 49

.

Adım Adım Çözüm

1

Identify the coefficients of the quadratic equation

3x^2 + 5x - 2 = 0

in the standard form

ax^2 + bx + c = 0

.

a = 3

,

b = 5

, and

c = -2

To use the discriminant formula, we must first extract the constant coefficients corresponding to each term.

2

Substitute the coefficients into the discriminant formula

D = b^2 - 4ac

.

D = 5^2 - 4(3)(-2) = 25 - (-24) = 25 + 24 = 49

Calculating the value of the discriminant provides the required solution.

Anahtar Kavram

Calculating the discriminant of a quadratic equation to determine the nature of its roots.

Soru 2034Soru →

What is the maximum integer value of $k$ that satisfies the inequality $8 - 5k > 28$ ?

Cevabı ve açıklamayı göster

Cevap: -5

Cevap

The maximum integer value that satisfies the inequality is

-5

.

Subtracting 8 from both sides of the inequality

8 - 5k > 28

yields

-5k > 20

. Dividing both sides of the inequality by

-5

and reversing the inequality sign results in

k < -4

. The largest integer strictly less than

-4

is

-5

.

Adım Adım Çözüm

1

Subtract 8 from both sides of the inequality.

-5k > 20

To isolate the term with the variable on the left side of the inequality.

2

Divide both sides by

-5

and reverse the inequality sign.

k < -4

Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality symbol.

3

Determine the largest integer strictly less than

-4

.

-5

Because the inequality is strict (

<

), the value of

k

cannot be equal to

-4

. The greatest integer less than

-4

is

-5

.

Anahtar Kavram

Solving linear inequalities and reversing the inequality sign when dividing by a negative number.

Soru 2035Soru →

Consider the functions $f(x) = \frac{x+3}{x-1}$ , where $x \neq 1$ , and $g(x) = x^2 - x - 4$ . If $x$ is an integer such that the composite function evaluation $g(f(g(x))) = 16$ , what is the product of all such integer values of $x$ ?

Cevabı ve açıklamayı göster

Cevap: -6

Cevap

The product of all integer values of

x

that satisfy the equation is

-6

.

Solving the nested composite equation

g(f(g(x))) = 16

systematically yields the integer solutions

x = 3

and

x = -2

. The product of these values is

-6

.

Adım Adım Çözüm

1

Set

u = f(g(x))

and solve the outer quadratic equation

g(u) = 16

.

u = 5

or

u = -4

This simplifies the nested composition into a single-variable quadratic equation.

2

Set

v = g(x)

and solve the rational equations

f(v) = 5

and

f(v) = -4

.

v = 2

or

v = 0.2

This determines the required outputs of the function

g(x)

that will satisfy the composite equation.

3

Solve the quadratic equations

g(x) = 2

and

g(x) = 0.2

for

x

.

x = 3

,

x = -2

, or

x = \frac{5 \pm \sqrt{445}}{10}

This finds all real values of

x

that satisfy the composite equation.

4

Identify the integer values from the solution set and compute their product.

3 \times (-2) = -6

The question specifically requests the product of the integer values of

x

.

Anahtar Kavram

Function composition and multi-step equation solving

Soru 2036Soru →

For a constant $k$ , the quadratic equation $\frac{1}{3}x^2 - \frac{k}{6}x + (k - 10) = 0$ has two real roots, $r_1$ and $r_2$ . If the sum of the reciprocals of the roots, $\frac{1}{r_1} + \frac{1}{r_2}$ , is equal to $-\frac{1}{2}$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 7.5

Cevap

The value of

k

is

7.5

.

By writing the sum of the reciprocals in terms of the sum and product of the roots, we find that

\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2}

. Substituting the values from Vieta's formulas (

r_1 + r_2 = \frac{k}{2}

and

r_1 r_2 = 3(k - 10)

) yields

\frac{k}{6(k - 10)} = -\frac{1}{2}

. Solving this equation for

k

results in

k = 7.5

.

Adım Adım Çözüm

1

Identify the quadratic coefficients in terms of the constant

k

.

a = \frac{1}{3}

,

b = -\frac{k}{6}

, and

c = k - 10

Applying Vieta's formulas requires the coefficients from the standard form

ax^2 + bx + c = 0

.

2

Determine the sum and the product of the roots using Vieta's formulas.

r_1 + r_2 = \frac{k}{2}

and

r_1 r_2 = 3(k - 10)

Vieta's formulas state that the sum of the roots is

-\frac{b}{a}

and the product of the roots is

\frac{c}{a}

.

3

Express the sum of the reciprocals of the roots in terms of

k

.

\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2} = \frac{k}{6(k - 10)}

Finding a common denominator allows the sum of the reciprocals to be written as the ratio of the sum of the roots to the product of the roots.

4

Equate the expression to

-\frac{1}{2}

and solve the resulting equation for

k

.

k = 7.5

Solving

\frac{k}{6(k - 10)} = -\frac{1}{2}

gives

2k = -6(k - 10) \implies 8k = 60 \implies k = 7.5

.

Anahtar Kavram

Vieta's Formulas and Algebraic Relationships of Roots

Tahmini Süre:2m 30s

Soru 2037Soru →

For the opening night of a school play, a total of $320$ tickets were sold, raising a total of $\$2{}140$ in ticket sales. Student tickets were sold for $\$5$ each, and adult tickets were sold for $\$8$ each. How many adult tickets were sold?

Cevabı ve açıklamayı göster

Cevap: 180

Cevap

180

The correct answer is

180

. By setting up a single-variable linear equation where

a

represents the number of adult tickets, the problem translates to

8a + 5(320 - a) = 2140

. Simplifying this yields

3a + 1600 = 2140

, which simplifies to

3a = 540

and results in

a = 180

.

Adım Adım Çözüm

1

Define variables for the unknown quantities based on the given total.

Let

a

represent the number of adult tickets sold. The number of student tickets sold is represented as

320 - a

.

Since the total number of tickets is

320

, subtracting the number of adult tickets from the total yields the number of student tickets.

2

Construct a linear equation using the ticket prices and the total revenue.

8a + 5(320 - a) = 2140

Multiplying the quantity of each ticket type by its respective price (

8

dollars for adult and

5

dollars for student) yields the total ticket sales revenue of

2,140

dollars.

3

Solve the equation for the variable

a

.

8a + 1600 - 5a = 2140 \implies 3a + 1600 = 2140 \implies 3a = 540 \implies a = 180

Apply the distributive property, group like terms, isolate the variable term, and divide to solve for the number of adult tickets.

Anahtar Kavram

Translating word problems into a single-variable linear equation and solving for the unknown quantity.

Soru 2038Soru →

If $\log_5 x = 3$ , what is the value of $x$ ?

8

15

25

125

243

Cevabı ve açıklamayı göster

Cevap: 125

Cevap

125

To solve the logarithmic equation

\log_5 x = 3

, we apply the fundamental definition of a logarithm. A logarithmic equation of the form

\log_b x = y

can be rewritten in exponential form as

b^y = x

. In this equation, the base

b

is 5 and the exponent

y

is 3. Rewriting gives

5^3 = x

. Evaluating

5^3

yields

5 \times 5 \times 5 = 125

. Therefore, the correct value of

x

is 125.

Adım Adım Çözüm

1

Apply the definition of a logarithm to rewrite the logarithmic equation in its equivalent exponential form.

5^3 = x

By definition,

\log_b x = y

is equivalent to

b^y = x

, where

b

is the base,

y

is the exponent, and

x

is the argument.

2

Evaluate the exponential expression

5^3

to solve for

x

.

x = 125

Cubing 5 means multiplying it by itself three times:

5 \times 5 \times 5 = 125

.

Anahtar Kavram

Definition of Logarithms

Soru 2039Soru →

The table below shows some values of the functions $f$ and $g$ for several integer values of $x$ .

$x$	$f(x)$	$g(x)$
$-2$	$3$	$1$
$-1$	$4$	$-2$
$0$	$-1$	$3$
$1$	$2$	$0$
$2$	$-2$	$-1$
$3$	$1$	$2$

What is the value of $f(f(3)) + g(g(-1))$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of the expression is

3

.

To evaluate

f(f(3)) + g(g(-1))

, we start by evaluating the innermost expressions. Looking at the table for

x = 3

, we find

f(3) = 1

. Looking at the table for

x = -1

, we find

g(-1) = -2

. Substituting these values into the outer functions gives

f(1) + g(-2)

. Using the table again, we look up

x = 1

to find

f(1) = 2

, and look up

x = -2

to find

g(-2) = 1

. Adding these two results yields

2 + 1 = 3

.

Adım Adım Çözüm

1

Find the inner function values from the given table.

f(3) = 1

and

g(-1) = -2

Before evaluating the composite functions, we must find the outputs of the innermost functions.

2

Evaluate the outer functions using the results from the first step.

f(f(3)) = f(1) = 2

and

g(g(-1)) = g(-2) = 1

Substitute the inner outputs into the outer functions and look up the new inputs in the table.

3

Add the two resulting values together.

2 + 1 = 3

Combine the two terms to find the total sum requested by the question.

Anahtar Kavram

Evaluating composite functions using tables of values

Tahmini Süre:1m 0s

Soru 2040Soru →

A company sells two types of gift baskets: Standard and Premium. The sales at the company's North and South branches are represented by matrix $S$ :

S = \begin{bmatrix} 12 & 8 \\ 15 & 10 \end{bmatrix}

where the rows represent the North and South branches, respectively, and the columns represent the number of Standard and Premium baskets sold, respectively. The cost of the items inside each basket is represented by matrix $C$ :

C = \begin{bmatrix} x & 24 \\ 30 & y \end{bmatrix}

where the rows represent Standard and Premium baskets, respectively, and the columns represent the cost of food items and the cost of gift items (in dollars), respectively. If the total food item cost at the North branch is $600$ and the total gift item cost at the South branch is $760$ , what is the value of $x + y$ ?

Cevabı ve açıklamayı göster

Cevap: 70

Cevap

The value of

x + y

is 70.

To find the total costs, the sales matrix

S

is multiplied by the unit cost matrix

C

. The product matrix

R = SC

is a

2 \times 2

matrix where the entry in Row 1, Column 1 represents the total food cost at the North branch, and the entry in Row 2, Column 2 represents the total gift cost at the South branch. Setting up the equations

12x + 240 = 600

and

360 + 10y = 760

yields

x = 30

and

y = 40

, which sum to 70.

Adım Adım Çözüm

1

Set up the matrix multiplication

R = SC

to represent the total costs.

R = \begin{bmatrix} 12x + 240 & 288 + 8y \\ 15x + 300 & 360 + 10y \end{bmatrix}

The product of the sales matrix and the unit cost matrix yields the total cost matrix, where rows represent the branches and columns represent the cost categories.

2

Identify the expression for the total food item cost at the North branch and solve for

x

.

x = 30

The total food item cost at the North branch is the entry in Row 1, Column 1 of the product matrix, which is

12x + 8(30) = 12x + 240

. Setting this equal to

600

gives

12x + 240 = 600

, which simplifies to

12x = 360

, so

x = 30

.

3

Identify the expression for the total gift item cost at the South branch and solve for

y

.

y = 40

The total gift item cost at the South branch is the entry in Row 2, Column 2 of the product matrix, which is

15(24) + 10y = 360 + 10y

. Setting this equal to

760

gives

360 + 10y = 760

, which simplifies to

10y = 400

, so

y = 40

.

4

Calculate the sum of the variables

x

and

y

.

x + y = 70

Substitute

x = 30

and

y = 40

into the expression

x + y

to find the final answer.

Anahtar Kavram

Matrix multiplication and translating real-world scenarios into matrix equations.

Tahmini Süre:3m 0s

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