Tüm alıştırma soruları

When the expression $5(2x - 3y) - 3(x - 4y)$ is simplified, what is the coefficient of $y$ ?

Cevabı ve açıklamayı göster

Cevap: -3

Cevap

The coefficient of

y

is

-3

.

Distributing

5

to

(2x - 3y)

yields

10x - 15y

. Distributing

-3

to

(x - 4y)

yields

-3x + 12y

. Combining the

y

terms gives

-15y + 12y = -3y

. Therefore, the coefficient of

y

is

-3

.

Adım Adım Çözüm

1

Distribute the coefficients outside the parentheses.

10x - 15y - 3x + 12y

To eliminate the parentheses so that like terms can be combined.

2

Group and combine the terms containing

y

.

-15y + 12y = -3y

To determine the final simplified term containing

y

and identify its coefficient.

Anahtar Kavram

Distributing terms (especially negative coefficients) and combining like terms.

Tahmini Süre:45s

Soru 62Soru →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined by $z = \frac{11 + 3i}{3 - i}$ . What is the real part of $z$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The real part of the complex number is 3.

By multiplying both the numerator and the denominator of

\frac{11 + 3i}{3 - i}

by the conjugate of the denominator,

3 + i

, we obtain

\frac{(11+3i)(3+i)}{(3-i)(3+i)} = \frac{33 + 11i + 9i + 3i^2}{9 - i^2} = \frac{30 + 20i}{10} = 3 + 2i

. The real part of this complex number is the term without

i

, which is 3.

Adım Adım Çözüm

1

Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator, which is

3 + i

.

z = \frac{(11 + 3i)(3 + i)}{(3 - i)(3 + i)}

To eliminate the imaginary unit from the denominator.

2

Expand and simplify the numerator using the distributive property and substituting

-1

for

i^2

.

(11 + 3i)(3 + i) = 33 + 11i + 9i + 3i^2 = 33 + 20i + 3(-1) = 30 + 20i

To combine the real and imaginary terms of the numerator.

3

Expand and simplify the denominator using the difference of squares property and substituting

-1

for

i^2

.

(3 - i)(3 + i) = 9 - i^2 = 9 - (-1) = 10

To find the real number denominator.

4

Divide each term in the simplified numerator by the denominator.

z = \frac{30 + 20i}{10} = 3 + 2i

To express the complex number in the standard form

a + bi

.

5

Extract the real part of the resulting complex number

3 + 2i

.

3

The real part of a complex number in the form

a + bi

is

a

.

Anahtar Kavram

Division of complex numbers using the complex conjugate

Alternatif Yöntem

Instead of simplifying the fraction directly, assume the resulting complex number is

x + yi

, where

x

represents the real part and

y

represents the imaginary part. We can set up the equation

x + yi = \frac{11 + 3i}{3 - i}

and multiply both sides by

3 - i

to get

(x + yi)(3 - i) = 11 + 3i

. Expanding the left side gives

(3x + y) + (3y - x)i = 11 + 3i

. Equating the real and imaginary parts yields a system of linear equations:

3x + y = 11

and

-x + 3y = 3

. Multiplying the second equation by 3 and adding it to the first equation gives

10y = 20

, which means

y = 2

. Substituting

y = 2

back into the first equation yields

3x + 2 = 11

, which simplifies to

3x = 9

, or

x = 3

. The real part is therefore 3.

Tahmini Süre:1m 30s

Soru 63Soru →

If $\sqrt{3x - 2} = 4$ , what is the value of $x$ ?

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Cevap: 6

Cevap

The value of

x

is 6.

Squaring both sides of the equation

\sqrt{3x - 2} = 4

eliminates the radical, leading to

3x - 2 = 16

. Adding 2 to both sides results in

3x = 18

. Dividing both sides by 3 gives

x = 6

. Substituting 6 back into the original equation yields

\sqrt{3(6) - 2} = \sqrt{18 - 2} = \sqrt{16} = 4

, which verifies that the solution is correct.

Adım Adım Çözüm

1

Square both sides of the equation to remove the radical.

3x - 2 = 16

Squaring a square root removes the radical sign since

(\sqrt{a})^2 = a

for non-negative values.

2

Add 2 to both sides of the equation.

3x = 18

Adding the constant term moves it to the other side to isolate the term containing the variable.

3

Divide both sides by 3.

x = 6

Dividing by the coefficient of

x

yields the final solution.

Anahtar Kavram

Solving basic radical equations by isolating the radical and squaring both sides.

Soru 64Soru →

For the imaginary unit $i$ , where $i^2 = -1$ , let $z$ be the complex number defined by $z = \frac{10 + ki}{2 - i}$ , where $k$ is a real constant. If the imaginary part of $z$ is $4$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

k

is

5

.

Multiplying the numerator and denominator of

z = \frac{10 + ki}{2 - i}

by the complex conjugate

2 + i

gives

z = \frac{(20 - k) + (10 + 2k)i}{5}

. The imaginary part is

\frac{10 + 2k}{5}

. Setting this expression equal to

4

and solving for

k

yields

k = 5

.

Adım Adım Çözüm

1

Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator, which is

2 + i

.

z = \frac{(10 + ki)(2 + i)}{(2 - i)(2 + i)}

To eliminate the imaginary unit from the denominator and express the complex number in standard form.

2

Expand both the numerator and the denominator, using the property

i^2 = -1

.

z = \frac{20 + 10i + 2ki + ki^2}{4 - i^2} = \frac{(20 - k) + (10 + 2k)i}{5}

To separate the real terms and imaginary terms in the numerator and simplify the denominator to a real number.

3

Express the complex number in standard form

a + bi

to identify the imaginary part.

z = \frac{20 - k}{5} + \left(\frac{10 + 2k}{5}\right)i

The imaginary part of a complex number is the coefficient of

i

, which is

\frac{10 + 2k}{5}

.

4

Set the imaginary part equal to

4

and solve the linear equation for

k

.

\frac{10 + 2k}{5} = 4 \implies 10 + 2k = 20 \implies 2k = 10 \implies k = 5

To find the specific value of the constant

k

that makes the imaginary part of

z

equal to

4

.

Anahtar Kavram

Rationalizing complex numbers and identifying real and imaginary components

Soru 65Soru →

The quadratic equation $x^2 - 4x - 12 = 0$ has two real solutions. What is the value of the positive solution to this equation?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The positive solution to the equation is

6

.

Factoring the quadratic trinomial

x^2 - 4x - 12 = 0

yields

(x - 6)(x + 2) = 0

. Setting the individual binomial factors to zero gives the solutions

x = 6

and

x = -2

. The positive solution among these is

6

.

Adım Adım Çözüm

1

Factor the quadratic equation

(x - 6)(x + 2) = 0

Factoring the trinomial

x^2 - 4x - 12

requires finding two integers whose product is

-12

and whose sum is

-4

. These numbers are

-6

and

2

.

2

Apply the zero product property

x - 6 = 0

or

x + 2 = 0

If the product of two factors is equal to zero, then at least one of the individual factors must equal zero.

3

Solve for the variable and identify the positive root

x = 6

and

x = -2

Solving the linear equations yields

x = 6

and

x = -2

. Since the question asks for the positive solution, we select

6

.

Anahtar Kavram

Solving quadratic equations by factoring

Soru 66Soru →

The trinomial $2x^2 + 7x + 3$ can be factored into the product of two binomials of the form $(2x + a)(x + b)$ , where $a$ and $b$ are integers. What is the value of the expression $a + 2b$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of the expression

a + 2b

is 7.

Expanding the factored template

(2x + a)(x + b)

yields

2x^2 + (a + 2b)x + ab

. Comparing this to the given expression

2x^2 + 7x + 3

, the coefficient of

x

on the left side is

a + 2b

, and on the right side is 7. Therefore,

a + 2b = 7

. Alternatively, factoring

2x^2 + 7x + 3

yields

(2x + 1)(x + 3)

, where

a = 1

and

b = 3

. Substituting these integers into

a + 2b

gives

1 + 2(3) = 7

.

Adım Adım Çözüm

1

Expand the expression

(2x + a)(x + b)

using the FOIL method.

2x^2 + 2bx + ax + ab = 2x^2 + (a + 2b)x + ab

Expanding the template allows direct comparison of its coefficients with the given trinomial.

2

Equate the coefficients of the expanded template to the given trinomial

2x^2 + 7x + 3

.

a + 2b = 7

and

ab = 3

For the two polynomial expressions to be equivalent for all values of

x

, their corresponding coefficients must be equal.

3

Identify the requested value directly from the system of equations.

7

The question asks for the value of

a + 2b

, which is precisely the coefficient of the linear

x

term.

Anahtar Kavram

Factoring quadratic trinomials with a leading coefficient greater than 1

Alternatif Yöntem

Factor the trinomial

2x^2 + 7x + 3

using the AC method: multiply the leading coefficient (2) and the constant term (3) to get 6. Find two numbers that multiply to 6 and add to 7, which are 6 and 1. Rewrite the middle term:

2x^2 + 6x + x + 3

. Factor by grouping:

2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)

. Compare this to

(2x + a)(x + b)

to find

a = 1

and

b = 3

, then compute

a + 2b = 1 + 2(3) = 7

.

Tahmini Süre:45s

Soru 67Soru →

For all real values of $x$ and $y$ , the expression $3x(x - 2y) - 2(x^2 - 4xy + y^2) - x^2$ can be written in the form $axy + by^2$ , where $a$ and $b$ are constants. What is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

a

is 2.

The value of

a

is 2 because distributing

3x(x - 2y)

yields

3x^2 - 6xy

, and distributing

-2(x^2 - 4xy + y^2)

yields

-2x^2 + 8xy - 2y^2

. Combining these with the

-x^2

term yields

(3-2-1)x^2 + (-6+8)xy - 2y^2 = 2xy - 2y^2

. Comparing this to

axy + by^2

shows that

a

, the coefficient of the

xy

term, is 2.

Adım Adım Çözüm

1

Distribute

3x

across the first parenthetical expression

(x - 2y)

3x^2 - 6xy

To clear the first set of parentheses by multiplying

3x

by each term inside.

2

Distribute

-2

across the second parenthetical expression

(x^2 - 4xy + y^2)

-2x^2 + 8xy - 2y^2

To clear the second set of parentheses. Note that multiplying

-2

by

-4xy

yields a positive term

+8xy

due to the sign rules.

3

Write the full expression and group like terms

(3x^2 - 2x^2 - x^2) + (-6xy + 8xy) - 2y^2

To group terms with identical variable parts so they can be combined.

4

Combine the coefficients of the grouped terms

2xy - 2y^2

Simplifying the groups:

3-2-1=0

for the

x^2

terms, and

-6+8=2

for the

xy

terms.

5

Compare the simplified expression to the form

axy + by^2

to find the coefficient

a

a = 2

The coefficient of the

xy

term is

2

, which corresponds to

a

in the target expression.

Anahtar Kavram

Simplifying Expressions and Combining Like Terms

Tahmini Süre:1m 30s

Soru 68Soru →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $w$ is defined as $w = \frac{6 + 4i}{2i}$ . What is the imaginary part of $w$ ?

Cevabı ve açıklamayı göster

Cevap: -3

Cevap

The imaginary part of

w

is

-3

.

Dividing each term in the numerator of

\frac{6 + 4i}{2i}

by the denominator

2i

yields

\frac{6}{2i} + \frac{4i}{2i}

, which simplifies to

\frac{3}{i} + 2

. Since

i^2 = -1

, the term

\frac{3}{i}

can be rationalized to

-3i

. Thus, the complex number in standard form is

2 - 3i

. The imaginary part is the real coefficient of

i

, which is

-3

.

Adım Adım Çözüm

1

Divide each term in the numerator by the denominator.

w = \frac{6}{2i} + \frac{4i}{2i}

This separates the quotient into two simpler terms that can be simplified individually.

2

Simplify both terms.

w = \frac{3}{i} + 2

Reduce the fractions by dividing out common factors in both the numerators and the denominators.

3

Rationalize the denominator of the imaginary term.

\frac{3}{i} \cdot \frac{i}{i} = \frac{3i}{i^2} = \frac{3i}{-1} = -3i

Multiply the numerator and denominator by

i

to eliminate the imaginary unit from the denominator, using the property

i^2 = -1

.

4

Combine the real and imaginary parts into standard form

a + bi

.

w = 2 - 3i

Group the real constant and the simplified imaginary term together.

5

Identify the imaginary part of the complex number.

-3

The imaginary part of a complex number

a + bi

is the real coefficient

b

of the imaginary unit

i

.

Anahtar Kavram

Simplifying a quotient of complex numbers by dividing by a pure imaginary number.

Soru 69Soru →

If $x$ is a real number that satisfies the equation $\sqrt{2x + 7} + \sqrt{x + 3} = 1$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: -3

Cevap

The only real solution to the equation is

-3

.

The value

-3

is the only real number that satisfies the original equation. Substituting

-3

back into the original equation yields

\sqrt{2(-3) + 7} + \sqrt{-3 + 3} = \sqrt{1} + 0 = 1

, which is true.

Adım Adım Çözüm

1

Isolate the first radical term.

\sqrt{2x + 7} = 1 - \sqrt{x + 3}

This allows for squaring both sides to eliminate one radical.

2

Square both sides and simplify.

2x + 7 = x + 4 - 2\sqrt{x + 3}

Squaring removes the radical on the left side, though it creates a middle term on the right side.

3

Isolate the remaining radical term.

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

Grouping the non-radical terms on one side prepares the equation for a second squaring step.

4

Square both sides again to eliminate the remaining radical.

x^2 + 6x + 9 = 4(x + 3)

Squaring both sides eliminates the radical completely, converting the expression into a polynomial equation.

5

Solve the quadratic equation.

x = -3

and

x = 1

Rearranging to

x^2 + 2x - 3 = 0

and factoring as

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

gives the candidate solutions.

6

Substitute candidates back into the original equation to check for extraneous solutions.

The only valid solution is

x = -3

.

Substituting

x = 1

yields

5 = 1

(invalid), while substituting

x = -3

yields

1 = 1

(valid).

Anahtar Kavram

Solving radical equations by isolating radicals and squaring, then testing for extraneous solutions.

Soru 70Soru →

If the expression $x^4 - 8x^2 + 16 - 9y^2$ is factored completely over the integers, the product of the factors can be written as $(x^2 - a - by)(x^2 - c + dy)$ , where $a$ , $b$ , $c$ , and $d$ are positive integers. What is the value of $a + b + c + d$ ?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

14

By grouping the first three terms, the expression

x^4 - 8x^2 + 16

is recognized as

(x^2 - 4)^2

. Substituting this back into the original expression gives

(x^2 - 4)^2 - (3y)^2

. Applying the difference of squares identity, this factors into

(x^2 - 4 - 3y)(x^2 - 4 + 3y)

. Comparing this result to

(x^2 - a - by)(x^2 - c + dy)

where

a, b, c, d

are positive integers yields

a = 4

,

b = 3

,

c = 4

, and

d = 3

. Summing these values gives

4 + 3 + 4 + 3 = 14

.

Adım Adım Çözüm

1

Group the first three terms of the polynomial.

x^4 - 8x^2 + 16 = (x^2 - 4)^2

To recognize the perfect square trinomial structure in terms of

x^2

.

2

Rewrite the original expression using the grouped terms.

(x^2 - 4)^2 - 9y^2 = (x^2 - 4)^2 - (3y)^2

To express the polynomial as a difference of squares.

3

Factor the expression using the difference of squares formula

A^2 - B^2 = (A - B)(A + B)

.

(x^2 - 4 - 3y)(x^2 - 4 + 3y)

To obtain the completely factored form over the integers.

4

Compare the factored expression to the given template

(x^2 - a - by)(x^2 - c + dy)

where

a, b, c, d

are positive integers.

a = 4

,

b = 3

,

c = 4

,

d = 3

To identify the values of the constants that satisfy the positivity constraint.

5

Calculate the sum of the identified values.

a + b + c + d = 4 + 3 + 4 + 3 = 14

To answer the question.

Anahtar Kavram

Factoring by grouping and the difference of squares

Soru 71Soru →

What is the sum of all real values of

x

that satisfy the equation

\frac{x^2 - 3x}{x^2 - 3x + 2} + \frac{x^2 - 3x - 1}{x^2 - 3x - 3} = \frac{11}{3}

?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The sum of all real values of

x

that satisfy the equation is

3

.

Substituting

y = x^2 - 3x

simplifies the original rational equation into the quadratic form

y^2 - y - 12 = 0

. Solving for

y

yields the values

4

and

-3

. Substituting back

x^2 - 3x

for

y

produces two quadratic equations. The first,

x^2 - 3x - 4 = 0

, has real solutions of

4

and

-1

. The second,

x^2 - 3x + 3 = 0

, has a negative discriminant and produces no real solutions. Summing the valid real solutions gives

4 + (-1) = 3

.

Adım Adım Çözüm

1

Substitute

y = x^2 - 3x

into the equation to simplify the rational terms.

\frac{y}{y+2} + \frac{y-1}{y-3} = \frac{11}{3}

Using a temporary variable simplifies the algebraic manipulation of the rational expressions.

2

Multiply the entire equation by the least common denominator,

3(y+2)(y-3)

, to eliminate all fractions.

3y(y-3) + 3(y-1)(y+2) = 11(y+2)(y-3)

, where

y \neq -2

and

y \neq 3

This clears the denominators so the equation can be solved as a polynomial.

3

Expand the terms on both sides of the equation.

(3y^2 - 9y) + (3y^2 + 3y - 6) = 11y^2 - 11y - 66

Expanding allows for combining like terms.

4

Combine like terms and move all terms to one side of the equation to set it equal to zero.

5y^2 - 5y - 60 = 0

This sets up the expression in the standard quadratic form

ay^2 + by + c = 0

.

5

Divide the entire quadratic equation by its greatest common factor,

5

, and factor the resulting expression.

y^2 - y - 12 = 0 \implies (y-4)(y+3) = 0 \implies y = 4 \text{ or } y = -3

Factoring solves for the possible values of the substituted variable

y

.

6

Substitute back

y = x^2 - 3x

for each case and solve the resulting quadratic equations for

x

.

For

y = 4

:

x^2 - 3x - 4 = 0 \implies (x-4)(x+1) = 0 \implies x = 4 \text{ or } x = -1

.
For

y = -3

:

x^2 - 3x + 3 = 0

. The discriminant is

(-3)^2 - 4(1)(3) = -3 < 0

, which means there are no real solutions.

This determines the real values of

x

that solve the original equation.

7

Verify that neither solution makes the original denominators zero, and add the valid real solutions.

4 + (-1) = 3

Since both

x = 4

and

x = -1

result in non-zero denominators, both are valid real solutions. Their sum is

3

.

Anahtar Kavram

Solving rational equations by utilizing algebraic substitution to reduce complexity and analyzing quadratic equations for real solutions.

Soru 72Soru →

When the expression $-(p - 2q)^3 - p(p - 3q)(2p + q) + 3q^2(p - q)$ is completely simplified by combining like terms, what is the coefficient of $p^2q$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The coefficient of

p^2q

in the fully simplified expression is 11.

Expanding the three components of the expression yields:

-(p - 2q)^3 = -p^3 + 6p^2q - 12pq^2 + 8q^3

;

-p(p - 3q)(2p + q) = -2p^3 + 5p^2q + 3pq^2

; and

3q^2(p - q) = 3pq^2 - 3q^3

. Combining these terms gives the simplified polynomial

-3p^3 + 11p^2q - 6pq^2 + 5q^3

. The coefficient of

p^2q

is 11.

Adım Adım Çözüm

1

Expand and negate the term

-(p - 2q)^3

-p^3 + 6p^2q - 12pq^2 + 8q^3

Using the binomial theorem expansion for

(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

and distributing the negative sign.

2

Multiply and distribute

-p(p - 3q)(2p + q)

-2p^3 + 5p^2q + 3pq^2

First multiply the binomials

(p - 3q)(2p + q) = 2p^2 - 5pq - 3q^2

, and then multiply each term by

-p

.

3

Distribute

3q^2(p - q)

3pq^2 - 3q^3

Multiply

3q^2

by both terms inside the binomial.

4

Combine the like terms of

p^2q

11

Identify and sum all terms containing

p^2q

:

6p^2q + 5p^2q = 11p^2q

.

Anahtar Kavram

Simplifying algebraic expressions by expanding polynomials, applying the distributive property with negative signs, and combining like terms.

Soru 73Soru →

For the imaginary unit $i$ , where $i^2 = -1$ , what is the real part of the complex number $z = (2 - i)^3 + 9i$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The real part of the complex number is 2.

Expanding the expression

(2 - i)^3

yields

2 - 11i

. Adding

9i

gives

2 - 2i

. The real part of this complex number is the term without

i

, which is 2.

Adım Adım Çözüm

1

Expand the squared binomial

(2 - i)^2

.

3 - 4i

Apply the identity

(a - b)^2 = a^2 - 2ab + b^2

and substitute

i^2 = -1

.

2

Multiply the result of the square by

(2 - i)

to calculate

(2 - i)^3

.

2 - 11i

Distribute the terms

(3 - 4i)(2 - i) = 6 - 3i - 8i + 4i^2

and substitute

i^2 = -1

.

3

Add

9i

to the simplified cube to find the complex number

z

.

2 - 2i

Combine the imaginary components:

-11i + 9i = -2i

.

4

Identify the real part of

z

.

2

The real part of a complex number

a + bi

is

a

.

Anahtar Kavram

Expanding complex binomials and simplifying powers of the imaginary unit

Soru 74Soru →

For all real values of $x$ and $y$ , the expression $4x^2 - 9y^2 - 12y - 4$ can be factored into the form $(2x + ay + b)(2x - cy - d)$ , where $a, b, c$ , and $d$ are positive integers. What is the value of $a + b + c + d$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The value of

a + b + c + d

is 10.

Grouping the

y

terms gives

4x^2 - (9y^2 + 12y + 4)

. Factoring the quadratic within the parentheses gives

4x^2 - (3y + 2)^2

. Applying the difference of squares identity

(u^2 - v^2) = (u + v)(u - v)

leads to

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

. Matching this expression to the template

(2x + ay + b)(2x - cy - d)

reveals that

a = 3

,

b = 2

,

c = 3

, and

d = 2

, all of which are positive integers. The sum of these values is

3 + 2 + 3 + 2 = 10

.

Adım Adım Çözüm

1

Group the terms containing

y

4x^2 - (9y^2 + 12y + 4)

Grouping the terms allows us to identify a perfect square trinomial pattern.

2

Factor the trinomial inside the parentheses

4x^2 - (3y + 2)^2

The expression

9y^2 + 12y + 4

is a perfect square trinomial of the form

(3y)^2 + 2(3y)(2) + 2^2

.

3

Apply the difference of squares identity

(2x + (3y + 2))(2x - (3y + 2))

The expression is in the form

u^2 - v^2

, where

u = 2x

and

v = 3y + 2

. Using

u^2 - v^2 = (u + v)(u - v)

factors the expression.

4

Simplify the factored binomials by distributing signs

(2x + 3y + 2)(2x - 3y - 2)

Simplifying the expressions removes inner parentheses, allowing comparison with the target template.

5

Compare with the template

(2x + ay + b)(2x - cy - d)

to find the constants

a = 3

,

b = 2

,

c = 3

,

d = 2

Matching the terms directly gives

ay = 3y \Rightarrow a = 3

,

b = 2

,

-cy = -3y \Rightarrow c = 3

, and

-d = -2 \Rightarrow d = 2

.

6

Calculate the sum of the positive integers

10

The question asks for the value of

a + b + c + d

, which is

3 + 2 + 3 + 2 = 10

.

Anahtar Kavram

Factoring polynomials using grouping, perfect square trinomials, and the difference of squares identity

Tahmini Süre:2m 0s

Soru 75Soru →

When the expression $2x(x^2 - 3xy + 2y^2) - (x - 2y)^3 - y(x^2 - 5xy + 4y^2)$ is fully simplified by combining like terms, what is the coefficient of the $xy^2$ term?

Cevabı ve açıklamayı göster

Cevap: -3

Cevap

The coefficient of the

xy^2

term is

-3

.

Expanding the terms of the expression yields

2x^3 - 6x^2y + 4xy^2

,

-x^3 + 6x^2y - 12xy^2 + 8y^3

, and

-x^2y + 5xy^2 - 4y^3

. Summing the coefficients of the

xy^2

terms gives

4 - 12 + 5 = -3

. Thus, the coefficient of

xy^2

is

-3

.

Adım Adım Çözüm

1

Expand the first term of the expression:

2x(x^2 - 3xy + 2y^2)

2x^3 - 6x^2y + 4xy^2

Distribute the monomial

2x

to each term inside the parentheses:

2x \cdot x^2 = 2x^3

,

2x \cdot (-3xy) = -6x^2y

, and

2x \cdot 2y^2 = 4xy^2

.

2

Expand the cubed binomial

(x - 2y)^3

and distribute the negative sign

-x^3 + 6x^2y - 12xy^2 + 8y^3

First, expand the binomial

(x - 2y)^3 = x^3 - 3(x^2)(2y) + 3(x)(2y)^2 - (2y)^3 = x^3 - 6x^2y + 12xy^2 - 8y^3

. Then, distribute the negative sign to all terms inside the parentheses.

3

Expand the third term:

-y(x^2 - 5xy + 4y^2)

-x^2y + 5xy^2 - 4y^3

Distribute the negative monomial

-y

to each term inside the parentheses:

-y \cdot x^2 = -x^2y

,

-y \cdot (-5xy) = 5xy^2

, and

-y \cdot 4y^2 = -4y^3

.

4

Group and combine all like terms

x^3 - x^2y - 3xy^2 + 4y^3

Combine the coefficients of the corresponding terms:

(2x^3 - x^3) + (-6x^2y + 6x^2y - x^2y) + (4xy^2 - 12xy^2 + 5xy^2) + (8y^3 - 4y^3) = x^3 - x^2y - 3xy^2 + 4y^3

.

Anahtar Kavram

Simplifying Expressions and Combining Like Terms

Tahmini Süre:3m 0s

Soru 76Soru →

For a certain positive number $y$ , the product of $y$ and the quantity $2y + 5$ is equal to $12$ . What is the value of $y$ ?

Cevabı ve açıklamayı göster

Cevap: 1.5

Cevap

The positive value of

y

is

1.5

.

The correct answer is

1.5

. By translating the word problem, we obtain

y(2y + 5) = 12

. Expanding this gives

2y^2 + 5y = 12

, and subtracting

12

from both sides yields the standard quadratic equation

2y^2 + 5y - 12 = 0

. Factoring this expression gives

(2y - 3)(y + 4) = 0

. Setting the first factor to zero yields

y = 1.5

, which is positive and therefore satisfies the given condition.

Adım Adım Çözüm

1

Write the equation representing the relationship.

y(2y + 5) = 12

To translate the verbal description into an algebraic equation.

2

Distribute

y

and set the equation equal to zero.

2y^2 + 5y - 12 = 0

To put the quadratic equation into standard form

ay^2 + by + c = 0

so it can be factored.

3

Factor the quadratic equation.

(2y - 3)(y + 4) = 0

To find the factors that multiply to give the quadratic expression.

4

Solve for

y

and apply the constraint.

y = 1.5

Setting the factors to zero gives

y = 1.5

and

y = -4

. Since the problem states

y

is positive, we select the positive root.

Anahtar Kavram

Solving a non-monic quadratic equation by factoring after translating a verbal description into algebra.

Soru 77Soru →

Let the functions $f$ and $g$ be defined by $f(x) = |2x - 5|$ and $g(x) = \sqrt{x + 7}$ for all real numbers in their respective domains. If $a$ is a real number such that $(f \circ g)(a) = 3$ , what is the sum of all possible values of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The sum of all possible values of

a

is

3

.

The correct answer is

3

. The composition

(f \circ g)(a) = 3

translates to

f(g(a)) = |2\sqrt{a + 7} - 5| = 3

. This absolute value relation splits into two equations:

2\sqrt{a + 7} - 5 = 3

and

2\sqrt{a + 7} - 5 = -3

. Solving the first equation yields

\sqrt{a + 7} = 4 \Rightarrow a = 9

. Solving the second equation yields

\sqrt{a + 7} = 1 \Rightarrow a = -6

. Since both

9

and

-6

are greater than or equal to

-7

, they are within the domain of the radical function. The sum of these values is

9 + (-6) = 3

.

Adım Adım Çözüm

1

Express the composition

(f \circ g)(a)

in terms of

a

.

|2\sqrt{a + 7} - 5| = 3

By definition of function composition,

(f \circ g)(a) = f(g(a))

. Substituting

g(a) = \sqrt{a + 7}

into the expression for

f(x)

yields

f(g(a)) = |2g(a) - 5| = |2\sqrt{a + 7} - 5|

.

2

Set up the two algebraic cases to eliminate the absolute value.

2\sqrt{a + 7} - 5 = 3

or

2\sqrt{a + 7} - 5 = -3

An absolute value equation of the form

|u| = c

(where

c \geq 0

) has two possible cases:

u = c

or

u = -c

.

3

Solve the first equation case for

a

.

a = 9

Adding

5

to both sides of

2\sqrt{a + 7} - 5 = 3

gives

2\sqrt{a + 7} = 8

. Dividing by

2

gives

\sqrt{a + 7} = 4

. Squaring both sides yields

a + 7 = 16

, which gives

a = 9

.

4

Solve the second equation case for

a

.

a = -6

Adding

5

to both sides of

2\sqrt{a + 7} - 5 = -3

gives

2\sqrt{a + 7} = 2

. Dividing by

2

gives

\sqrt{a + 7} = 1

. Squaring both sides yields

a + 7 = 1

, which gives

a = -6

.

5

Verify domain constraints and sum the valid solutions.

3

Both

a = 9

and

a = -6

satisfy the domain requirement for

g(x) = \sqrt{x+7}

, which is

x \geq -7

. The sum of these two valid values is

9 + (-6) = 3

.

Anahtar Kavram

Evaluating and solving equations containing composite functions, absolute values, and radical functions.

Alternatif Yöntem

Instead of expanding the composition immediately, substitute a temporary variable

u = g(a) = \sqrt{a+7}

. The equation becomes

f(u) = 3 \Rightarrow |2u - 5| = 3

. Solve this simplified absolute value equation to get

2u - 5 = 3 \Rightarrow u = 4

, and

2u - 5 = -3 \Rightarrow u = 1

. Next, substitute back

\sqrt{a+7}

for

u

: solving

\sqrt{a+7} = 4

yields

a = 9

, and solving

\sqrt{a+7} = 1

yields

a = -6

. Summing these two solutions gives

9 + (-6) = 3

.

Tahmini Süre:2m 0s

Soru 78Soru →

A specialty coffee shop blends three types of coffee beans: Colombian ( $8.00$ dollars per pound), Ethiopian ( $11.00$ dollars per pound), and Sumatran ( $14.00$ dollars per pound). The shop manager wants to create a $50$ -pound blend that costs exactly $11.60$ dollars per pound. Due to supply constraints, the weight of the Ethiopian beans in the blend must be exactly $2$ pounds more than $\frac{1}{5}$ of the combined weight of the Colombian and Sumatran beans. What is the value of $2s - c$ , where $s$ is the amount of Sumatran beans, in pounds, and $c$ is the amount of Colombian beans, in pounds, in the blend?

Cevabı ve açıklamayı göster

Cevap: 35

Cevap

The correct answer is

35

.

The correct answer of

35

is found by translating the given constraints into three linear equations, solving for

e = 10

using substitution of the sum

c+s = 50 - e

, and then solving the resulting two-variable system to find

s = 25

and

c = 15

. Evaluating

2s - c

yields

2(25) - 15 = 35

.

Adım Adım Çözüm

1

Define variables and write the system of equations representing the constraints.

Let

c

,

e

, and

s

represent the weight in pounds of Colombian, Ethiopian, and Sumatran beans, respectively. The system is:
1)

c + e + s = 50

(total weight)
2)

8c + 11e + 14s = 50 \times 11.60 = 580

(total cost)
3)

e = \frac{1}{5}(c + s) + 2

(Ethiopian bean weight constraint)

This translates the word problem's conditions into algebraic expressions.

2

Solve for the variable

e

using substitution.

From equation (1), we have

c + s = 50 - e

. Substituting this expression into equation (3) yields:

e = \frac{1}{5}(50 - e) + 2

Multiply both sides by

5

:

5e = 50 - e + 10

6e = 60 \implies e = 10

Grouping

c + s

allows us to solve for

e

directly without dealing with three separate variable eliminations.

3

Substitute

e = 10

back into the first two equations to simplify the system to two variables.

Equation (1) becomes:

c + s = 40 \implies c = 40 - s

Equation (2) becomes:

8c + 11(10) + 14s = 580 \implies 8c + 14s = 470

This reduces the remaining problem to a standard system of two linear equations.

4

Solve for

c

and

s

by substitution.

Substitute

c = 40 - s

into the simplified cost equation:

8(40 - s) + 14s = 470

320 - 8s + 14s = 470

320 + 6s = 470 \implies 6s = 150 \implies s = 25

Then, find

c

:

c = 40 - 25 = 15

This gives the exact weight of both the Sumatran and Colombian beans.

5

Evaluate the expression

2s - c

requested by the prompt.

2s - c = 2(25) - 15 = 50 - 15 = 35

This satisfies the specific algebraic quantity requested in the question.

Anahtar Kavram

Translating word problems into systems of linear equations and solving them using algebraic substitution.

Soru 79Soru →

Let the functions $f$ and $g$ be defined for all real numbers by $f(x) = 2x - 3$ and $g(x) = x + 5$ . What is the value of $f(g(4))$ ?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The value of

f(g(4))

is

15

.

To evaluate the composite function

f(g(4))

, the inner function must be evaluated first. Substituting

4

into

g(x) = x + 5

gives

g(4) = 9

. Next, substitute this output of

9

into the outer function

f(x) = 2x - 3

, which gives

f(9) = 2(9) - 3 = 15

.

Adım Adım Çözüm

1

Evaluate the inner function

g(4)

g(4) = 9

Substitute

4

for

x

in the definition of

g(x) = x + 5

.

2

Evaluate the outer function

f(x)

at the output of the inner function

f(9) = 15

Substitute

9

(the value of

g(4)

) for

x

in the definition of

f(x) = 2x - 3

.

Anahtar Kavram

Function Composition and Evaluation

Tahmini Süre:45s

Soru 80Soru →

A rental car company charges a flat fee of $30$ dollars plus $0.20$ dollars per mile driven. If a customer's total rental bill is $56$ dollars, how many miles did they drive?

Cevabı ve açıklamayı göster

Cevap: 130

Cevap

130 miles

Subtracting the flat fee of

30

dollars from the total bill of

56

dollars leaves

26

dollars representing the cost of the miles driven. Dividing this remaining cost of

26

dollars by the rate of

0.20

dollars per mile yields a total of

130

miles driven.

Adım Adım Çözüm

1

Set up a linear equation for the total cost

30 + 0.20m = 56

The total cost of

56

dollars is the sum of the flat fee of

30

dollars and the variable cost of

0.20

dollars per mile multiplied by the number of miles

m

.

2

Subtract the flat fee from both sides of the equation

0.20m = 26

Isolating the variable term shows that the total amount spent on the mileage portion of the trip is

26

dollars.

3

Divide by the rate per mile to find the total miles

m = 130

Dividing the mileage portion of the cost by the cost per mile gives the total distance driven.

Anahtar Kavram

Translating and Solving Linear Word Problems

Cevap

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Alternatif Yöntem

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