Elementary Algebra Alıştırma Soruları — ACT

Soru 1Soru →

The quadratic expression $x^2 - 5x - 14$ can be factored completely into $(x + a)(x + b)$ , where $a$ and $b$ are integers and $a > b$ . What is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2

To factor

x^2 - 5x - 14

, we look for two integers that multiply to

-14

and add to

-5

. These integers are

-7

and

2

. Thus, the factored form is

(x - 7)(x + 2)

, which corresponds to

(x + a)(x + b)

where the two constant values are

-7

and

2

. Given the condition

a > b

, the larger value must be assigned to

a

. Since

2 > -7

, we find

a = 2

.

Adım Adım Çözüm

1

Find two integers that multiply to the constant term

-14

and add to the linear coefficient

-5

.

The two integers are

-7

and

2

.

Since

(-7) \times 2 = -14

and

-7 + 2 = -5

, these integers satisfy the requirements for factoring the quadratic trinomial.

2

Write the quadratic expression in its factored form

(x + p)(x + q)

.

(x - 7)(x + 2)

The quadratic expression

x^2 + Bx + C

factors into

(x + p)(x + q)

where

p

and

q

are the found integers.

3

Compare the factored form to the template

(x + a)(x + b)

under the condition

a > b

.

The set of constants is

\{-7, 2\}

. Since

2 > -7

, we assign

a = 2

and

b = -7

.

This satisfies the requirement that the integer

a

is strictly greater than the integer

b

.

Anahtar Kavram

Factoring quadratic trinomials with a leading coefficient of 1

Tahmini Süre:45s

Soru 2Soru →

Match each of the following quadratic equations to its correct set of real solutions.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The equation

2x(x + 1) = 12 - 3x

matches with the solution set

\{-4, \frac{3}{2}\}

; the equation

3x(x - 1) = 2(x + 4)

matches with the solution set

\{-1, \frac{8}{3}\}

; and the equation

4x(x - 2) = 5

matches with the solution set

\{-\frac{1}{2}, \frac{5}{2}\}

.

Each equation is correctly solved by first distributing, moving all terms to one side to set the equation to zero, factoring the resulting trinomial over the integers, and then applying the zero product property to find the corresponding solution set.

Adım Adım Çözüm

1

Rearrange the first equation

2x(x + 1) = 12 - 3x

into standard form

ax^2 + bx + c = 0

.

2x^2 + 5x - 12 = 0

Distributing the

2x

gives

2x^2 + 2x = 12 - 3x

. Adding

3x

and subtracting

12

from both sides moves all terms to one side.

2

Factor the rearranged first equation

2x^2 + 5x - 12 = 0

and solve for

x

.

x = -4

or

x = \frac{3}{2}

Finding two integers that multiply to

-24

and add to

5

gives

8

and

-3

. Splitting the middle term and factoring by grouping yields

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

. Setting each factor to zero gives the solutions.

3

Rearrange the second equation

3x(x - 1) = 2(x + 4)

into standard form

ax^2 + bx + c = 0

.

3x^2 - 5x - 8 = 0

Distributing on both sides gives

3x^2 - 3x = 2x + 8

. Subtracting

2x

and

8

from both sides sets the quadratic expression to zero.

4

Factor the rearranged second equation

3x^2 - 5x - 8 = 0

and solve for

x

.

x = -1

or

x = \frac{8}{3}

Finding two integers that multiply to

-24

and add to

-5

gives

-8

and

3

. Grouping terms gives

(3x - 8)(x + 1) = 0

. Setting the factors to zero gives the solutions.

5

Rearrange the third equation

4x(x - 2) = 5

into standard form

ax^2 + bx + c = 0

.

4x^2 - 8x - 5 = 0

Distributing the

4x

yields

4x^2 - 8x = 5

. Subtracting

5

from both sides sets the equation to zero.

6

Factor the rearranged third equation

4x^2 - 8x - 5 = 0

and solve for

x

.

x = -\frac{1}{2}

or

x = \frac{5}{2}

Finding two integers that multiply to

-20

and add to

-8

gives

-10

and

2

. Grouping terms gives

(2x + 1)(2x - 5) = 0

. Solving each linear factor for

x

provides the solutions.

Anahtar Kavram

Solving quadratic equations by rearranging them into standard form, factoring by grouping, and applying the zero product property.

Soru 3Soru →

The polynomial $x^2 + 8x + 15$ can be factored into the form $(x + a)(x + b)$ , where $a$ and $b$ are integers such that $a < b$ . What is the value of $2a + b$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The value of

2a + b

is 11.

Factoring the trinomial

x^2 + 8x + 15

gives

(x + 3)(x + 5)

. Since

a < b

, we must have

a = 3

and

b = 5

. Thus,

2a + b = 2(3) + 5 = 11

.

Adım Adım Çözüm

1

Factor the quadratic expression

x^2 + 8x + 15

.

(x + 3)(x + 5)

To factor the trinomial, we find two integers that multiply to the constant term 15 and add to the linear coefficient 8. The integers 3 and 5 satisfy these requirements.

2

Assign the values to

a

and

b

under the condition

a < b

.

a = 3

and

b = 5

Comparing

(x + 3)(x + 5)

to

(x + a)(x + b)

gives the values 3 and 5. The condition

a < b

dictates that the smaller value 3 goes to

a

and the larger value 5 goes to

b

.

3

Calculate the value of

2a + b

.

11

Substitute

a = 3

and

b = 5

into the expression:

2(3) + 5 = 6 + 5 = 11

.

Anahtar Kavram

Factoring quadratic trinomials

Soru 4Soru →

If $3x(x - 3) = 2(x + 3) - 12$ , what is the product of the two solutions to this equation?

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Cevap: 2

Cevap

The product of the two solutions to the equation is 2.

The correct answer is 2. Expanding both sides of the equation

3x(x - 3) = 2(x + 3) - 12

gives

3x^2 - 9x = 2x + 6 - 12

, which simplifies to

3x^2 - 9x = 2x - 6

. Subtracting

2x - 6

from both sides results in the standard form quadratic equation

3x^2 - 11x + 6 = 0

. Factoring this expression gives

(3x - 2)(x - 3) = 0

. Setting each factor to zero yields the solutions

x = \frac{2}{3}

and

x = 3

. Multiplying these solutions gives

\frac{2}{3} \times 3 = 2

. Alternatively, by Vieta's formulas, the product of the roots of a quadratic equation

ax^2 + bx + c = 0

is

\frac{c}{a}

, which directly gives

\frac{6}{3} = 2

.

Adım Adım Çözüm

1

Expand the expressions on both sides of the equation.

3x^2 - 9x = 2x + 6 - 12

, which simplifies to

3x^2 - 9x = 2x - 6

To prepare the equation for rearrangement into the standard quadratic form.

2

Move all terms to the left side of the equation to set it equal to zero.

3x^2 - 11x + 6 = 0

To write the quadratic equation in the standard form

ax^2 + bx + c = 0

.

3

Factor the quadratic trinomial by grouping.

(3x - 2)(x - 3) = 0

To break down the quadratic equation into linear factors that can be solved individually.

4

Set each linear factor to zero and solve for

x

.

x = \frac{2}{3}

and

x = 3

According to the zero product property, if a product of factors is zero, at least one factor must be zero.

5

Calculate the product of the two solutions.

\frac{2}{3} \times 3 = 2

To find the product of the solutions as requested by the question.

Anahtar Kavram

Solving quadratic equations by factoring after expanding and rearranging terms

Soru 5Soru →

A charity walkathon organizer pledges to donate a base amount of $\$150$ plus $\$2.50$ for every kilometer completed by each participant. On the day of the event, a participant completes a distance that is $3$ kilometers less than twice their training average distance. If the organizer's donation for this participant is $\$212.50$ , what is the participant's training average distance, in kilometers?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

The participant's training average distance is

14

kilometers.

The correct answer is

14

. To find this value, we write the donation relationship as

150 + 2.50d = 212.50

, where

d

is the distance completed. Solving for

d

gives

d = 25

kilometers. Next, we translate the phrase '3 kilometers less than twice their training average distance' into the expression

2A - 3

, where

A

is the training average. Equating this to the completed distance gives

2A - 3 = 25

. Solving for

A

yields

A = 14

.

Adım Adım Çözüm

1

Set up an equation for the total donation as a function of the distance completed,

d

, in kilometers.

150 + 2.50d = 212.50

The total donation is the sum of the flat base donation of

\$150

and the rate of

\$2.50

per kilometer completed.

2

Solve the equation for the completed distance,

d

.

d = 25

Subtracting

150

from both sides of the equation yields

2.50d = 62.50

. Dividing both sides by

2.50

yields

d = 25

.

3

Translate the relationship between the completed distance,

d

, and the training average distance,

A

, into an equation.

2A - 3 = 25

The phrase '3 kilometers less than twice their training average distance' translates mathematically to

2A - 3

. Since the completed distance is

25

kilometers, we set the expression equal to

25

.

4

Solve the equation for the training average distance,

A

.

A = 14

Adding

3

to both sides of the equation yields

2A = 28

. Dividing both sides by

2

yields

A = 14

.

Anahtar Kavram

Translating word problems into multi-step linear equations and solving for the unknown variable.

Tahmini Süre:1m 30s

Soru 6Soru →

If $x = -2$ , $y = 5$ , and $z = -\frac{1}{2}$ , what is the value of the algebraic expression $\frac{x^3 y + z^{-2}}{y - 2x}$ ?

Cevabı ve açıklamayı göster

Cevap: -4

Cevap

The value of the expression is -4.

Substituting

x = -2

,

y = 5

, and

z = -\frac{1}{2}

into the expression yields

x^3 y = (-2)^3(5) = -40

and

z^{-2} = \left(-\frac{1}{2}\right)^{-2} = 4

, making the numerator

-40 + 4 = -36

. Evaluating the denominator gives

y - 2x = 5 - 2(-2) = 9

. Dividing

-36

by

9

results in

-4

.

Adım Adım Çözüm

1

Evaluate the terms in the numerator individually.

x^3 y = (-2)^3(5) = -40

and

z^{-2} = \left(-\frac{1}{2}\right)^{-2} = 4

.

Negative bases raised to odd powers retain a negative sign, while negative exponents represent the reciprocal raised to a positive power.

2

Calculate the total numerator value.

-40 + 4 = -36

.

Summing the two evaluated terms gives the complete numerator.

3

Evaluate the denominator expression.

y - 2x = 5 - 2(-2) = 5 + 4 = 9

.

Subtracting a negative quantity is equivalent to adding its positive counterpart.

4

Divide the numerator by the denominator.

\frac{-36}{9} = -4.

Dividing a negative integer by a positive integer yields a negative quotient.

Anahtar Kavram

Evaluating Algebraic Expressions with Negative Integers and Negative Exponents

Soru 7Soru →

For all non-zero real numbers $a$ and $b$ , the expression $(a^2 b^k)^3$ is equivalent to $a^6 b^{15}$ . What is the value of the integer $k$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of the integer

k

is 5.

To find the value of

k

, we simplify the expression

(a^2 b^k)^3

using exponent rules. According to the power of a product property,

(xy)^z = x^z y^z

, so

(a^2 b^k)^3 = (a^2)^3 (b^k)^3

. Next, applying the power of a power property,

(x^y)^z = x^{yz}

, we get

a^{2 \cdot 3} b^{k \cdot 3} = a^6 b^{3k}

. Since this expression is equivalent to

a^6 b^{15}

, we set the exponents of

b

equal to each other:

3k = 15

. Dividing both sides by 3 yields

k = 5

.

Adım Adım Çözüm

1

Apply the power of a product rule to the expression

(a^2 b^k)^3

.

(a^2)^3 \cdot (b^k)^3

The power of a product rule states that

(xy)^z = x^z y^z

.

2

Apply the power of a power rule to simplify the exponents.

a^6 b^{3k}

The power of a power rule states that

(x^y)^z = x^{y \cdot z}

, so

(a^2)^3 = a^{2 \cdot 3} = a^6

and

(b^k)^3 = b^{3k}

.

3

Set the exponent of

b

in

a^6 b^{3k}

equal to the exponent of

b

in the equivalent expression

a^6 b^{15}

.

3k = 15

Since the expressions are equivalent for all non-zero real numbers, the exponents of like bases must be equal.

4

Solve the linear equation for

k

.

k = 5

Dividing both sides of

3k = 15

by 3 isolates the variable

k

.

Anahtar Kavram

Properties of exponents, specifically the power of a product rule

(xy)^z = x^z y^z

and the power of a power rule

(x^y)^z = x^{y \cdot z}

.

Tahmini Süre:45s

Soru 8Soru →

What is the value of $x$ that satisfies the equation $\frac{3}{4}(x - 8) = \frac{1}{2}x + 2$ ?

Cevabı ve açıklamayı göster

Cevap: 32

Cevap

The correct value of

x

is

32

.

Distributing

\frac{3}{4}

on the left side yields

\frac{3}{4}x - 6

. Subtracting

\frac{1}{2}x

(which is

\frac{2}{4}x

) from both sides gives

\frac{1}{4}x - 6 = 2

. Adding 6 to both sides gives

\frac{1}{4}x = 8

, and multiplying by 4 yields the correct value of

32

.

Adım Adım Çözüm

1

Distribute the fraction

\frac{3}{4}

to both terms inside the parentheses.

\frac{3}{4}x - 6 = \frac{1}{2}x + 2

To simplify the equation by removing the parentheses.

2

Subtract

\frac{1}{2}x

from both sides of the equation.

\frac{1}{4}x - 6 = 2

To collect all terms with the variable

x

on one side of the equation.

3

Add 6 to both sides of the equation.

\frac{1}{4}x = 8

To isolate the term containing the variable.

4

Multiply both sides of the equation by 4.

x = 32

To solve for

x

by eliminating the coefficient of

\frac{1}{4}

.

Anahtar Kavram

Solving linear equations with variables on both sides and fractional coefficients

Alternatif Yöntem

Multiply the entire equation by the least common denominator of the fractions, which is 4, to clear the fractions before solving:

4 \cdot [\frac{3}{4}(x - 8)] = 4 \cdot [\frac{1}{2}x + 2] \implies 3(x - 8) = 2x + 8

. Then distribute:

3x - 24 = 2x + 8

. Subtract

2x

from both sides:

x - 24 = 8

. Add 24 to both sides:

x = 32

.

Tahmini Süre:45s

Soru 9Soru →

If the expression $\frac{(p^3 q^{-2})^{-1}}{(p^{-1} q^2)^3}$ is simplified to the form $p^x q^y$ for all non-zero real numbers $p$ and $q$ , what is the value of $2x - y$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The correct answer is 4.

Applying the power of a power rule to the numerator yields

(p^3 q^{-2})^{-1} = p^{-3} q^2

. Applying the same rule to the denominator yields

(p^{-1} q^2)^3 = p^{-3} q^6

. Dividing the terms by subtracting exponents gives

p^{-3 - (-3)} q^{2-6} = p^0 q^{-4}

. Thus,

x = 0

and

y = -4

. Evaluating

2x - y

gives

2(0) - (-4) = 4

.

Adım Adım Çözüm

1

Simplify the numerator using the power of a power property, which states that

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

.

(p^3 q^{-2})^{-1} = p^{3 \cdot (-1)} q^{-2 \cdot (-1)} = p^{-3} q^2

This distributes the exponent of

-1

to both factors inside the parentheses in the numerator.

2

Simplify the denominator using the power of a power property, which states that

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

.

(p^{-1} q^2)^3 = p^{-1 \cdot 3} q^{2 \cdot 3} = p^{-3} q^6

This distributes the exponent of

3

to both factors inside the parentheses in the denominator.

3

Combine the simplified numerator and denominator using the quotient property of exponents, which states that

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

.

\frac{p^{-3} q^2}{p^{-3} q^6} = p^{-3 - (-3)} q^{2 - 6} = p^0 q^{-4}

This simplifies division by subtracting the exponent in the denominator from the exponent in the numerator for each base.

4

Identify the values of

x

and

y

from the simplified form

p^0 q^{-4}

, and evaluate the final expression

2x - y

.

x = 0

and

y = -4

, so

2(0) - (-4) = 4

This substitutes the values of the exponents into the target algebraic expression to find the final numerical answer.

Anahtar Kavram

Properties of exponents including power of a power and quotient properties

Alternatif Yöntem

Alternatively, you can write the terms with positive exponents first:

\frac{(p^3 q^{-2})^{-1}}{(p^{-1} q^2)^3} = \frac{(p^{-1} q^2)^3}{(p^3 q^{-2})^1} = \frac{p^{-3} q^6}{p^3 q^{-2}} = p^{-3-3} q^{6-(-2)} = p^{-6} q^8

, but this expression is equivalent to the original expression only if inverted properly. Direct distribution is less prone to inversion errors.

Tahmini Süre:1m 30s

Soru 10Soru →

Which of the following is the completely factored form of $x^2 - 64$ ?

(x - 8)^2

(x - 32)(x + 32)

(x - 8)(x + 8)

(x^2 - 8)(x^2 + 8)

(x - 16)(x + 4)

Cevabı ve açıklamayı göster

Cevap: $(x - 8)(x + 8)$

Cevap

(x - 8)(x + 8)

The polynomial

x^2 - 64

is a difference of two squares because

x^2

is the square of

x

and

64

is the square of

8

. Applying the difference of squares formula,

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

, with

a = x

and

b = 8

gives the factored form

(x - 8)(x + 8)

.

Adım Adım Çözüm

1

Identify the structure of the polynomial.

The expression

x^2 - 64

is a difference of two squares since

x^2

is

(x)^2

and

64

is

(8)^2

.

Recognizing the difference of squares allows the use of the algebraic identity

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

.

2

Apply the difference of squares identity with

a = x

and

b = 8

.

(x - 8)(x + 8)

Substituting

x

and

8

into the formula yields the factored form.

Anahtar Kavram

Factoring the difference of squares using the identity

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

Soru 11Soru →

Which of the following is equivalent to the expression $4x^2(x - 2y) - 3x(x^2 - 3xy) - (x^3 + xy^2)$ for all real values of $x$ and $y$ ?

x^2y + xy^2

-17x^2y - xy^2

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

x^2y - xy^2

-17x^2y + xy^2

Cevabı ve açıklamayı göster

Cevap: $x^2y - xy^2$

Cevap

The correct simplified expression is

x^2y - xy^2

.

Distributing the terms yields

4x^3 - 8x^2y - 3x^3 + 9x^2y - x^3 - xy^2

. Grouping and combining like terms results in

(4-3-1)x^3 + (-8+9)x^2y - xy^2 = x^2y - xy^2

.

Adım Adım Çözüm

1

Distribute the term

4x^2

to the terms inside the first set of parentheses:

4x^2(x - 2y)

.

4x^3 - 8x^2y

Multiplying a monomial by a binomial requires distributing the multiplier to each term inside.

2

Distribute the term

-3x

to the terms inside the second set of parentheses:

-3x(x^2 - 3xy)

.

-3x^3 + 9x^2y

Carefully apply the distributive property, remembering that multiplying a negative term by a negative term yields a positive term:

-3x \cdot (-3xy) = 9x^2y

.

3

Distribute the negative sign (or

-1

) to the terms inside the third set of parentheses:

-(x^3 + xy^2)

.

-x^3 - xy^2

Distributing the negative sign changes the signs of all terms inside the parentheses.

4

Combine all parts and group the like terms:

x^3

,

x^2y

, and

xy^2

.

x^2y - xy^2

Add the coefficients of the terms with the same variable parts:

(4 - 3 - 1)x^3 + (-8 + 9)x^2y - xy^2 = 0x^3 + x^2y - xy^2 = x^2y - xy^2

.

Anahtar Kavram

Simplifying algebraic expressions by distributing and combining like terms.

Tahmini Süre:1m 0s

Soru 12Soru →

What is the positive difference between the two real solutions to the quadratic equation $2x(3x - 5) = 3(x - 2)$ ?

\frac{1}{6}

1

\frac{5}{6}

\frac{11}{6}

\frac{13}{6}

Cevabı ve açıklamayı göster

Cevap: $\frac{5}{6}$

Cevap

The positive difference between the two solutions is

\frac{5}{6}

.

The correct positive difference is

\frac{5}{6}

because rearranging the equation

2x(3x - 5) = 3(x - 2)

yields

6x^2 - 13x + 6 = 0

. Factoring this equation gives

(2x - 3)(3x - 2) = 0

, which has the solutions

x = \frac{3}{2}

and

x = \frac{2}{3}

. Subtracting these values gives

\frac{3}{2} - \frac{2}{3} = \frac{9 - 4}{6} = \frac{5}{6}

.

Adım Adım Çözüm

1

Expand both sides of the equation.

6x^2 - 10x = 3x - 6

To begin solving, distribute the term

2x

on the left side and the constant

3

on the right side.

2

Rearrange the terms to set the quadratic equation to zero.

6x^2 - 13x + 6 = 0

Subtract

3x

and add

6

to both sides of the equation to write it in standard form

ax^2 + bx + c = 0

.

3

Factor the quadratic equation over the integers.

(2x - 3)(3x - 2) = 0

Find two binomials whose product is

6x^2 - 13x + 6

. We search for two numbers that multiply to

36

(from

6 \times 6

) and sum to

-13

, which are

-9

and

-4

, allowing factoring by grouping.

4

Solve for the roots of the equation.

x = \frac{3}{2}

or

x = \frac{2}{3}

Set each linear factor equal to zero using the Zero Product Property:

2x - 3 = 0 \implies x = \frac{3}{2}

and

3x - 2 = 0 \implies x = \frac{2}{3}

.

5

Calculate the positive difference between the two solutions.

\frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} = \frac{5}{6}

Subtract the smaller solution from the larger solution to find the positive difference.

Anahtar Kavram

Solving quadratic equations by rearranging and factoring over integers.

Soru 13Soru →

For all real numbers $x$ and $y$ , which of the following is equivalent to the expression $(x - 2y)^2 - (x^2 - 5xy)$ ?

4y + xy

4y^2 - 9xy

4y^2 + 5xy

4y^2 + xy

4y^2 - 5xy

Cevabı ve açıklamayı göster

Cevap: $4y^2 + xy$

Cevap

The expression is equivalent to

4y^2 + xy

.

Expanding

(x - 2y)^2

yields

x^2 - 4xy + 4y^2

. Distributing the negative sign to

-(x^2 - 5xy)

gives

-x^2 + 5xy

. Combining the results yields

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

, which is equivalent to

4y^2 + xy

.

Adım Adım Çözüm

1

Expand the squared binomial

(x - 2y)^2

.

x^2 - 4xy + 4y^2

Apply the binomial squaring formula

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

to expand the expression.

2

Distribute the negative sign across the second parenthetical expression

-(x^2 - 5xy)

.

-x^2 + 5xy

Distributing the negative sign changes the signs of both terms inside the parentheses.

3

Combine like terms from the expanded parts.

xy + 4y^2

Group and add coefficients of the like terms:

(x^2 - x^2) + (-4xy + 5xy) + 4y^2 = 0 + xy + 4y^2

.

Anahtar Kavram

Simplifying Expressions and Combining Like Terms

Soru 14Soru →

Which of the following expressions represents the complete factorization of $2x^4 - 32$ ?

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

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

2(x - 2)(x + 2)(x + 2)^2

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

2(x - 4)(x + 4)

Cevabı ve açıklamayı göster

Cevap: $2(x - 2)(x + 2)(x^2 + 4)$

Cevap

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

The correct answer is found by first factoring out the greatest common factor of

2

, resulting in

2(x^4 - 16)

. Next, recognize that

x^4 - 16

is a difference of squares,

(x^2)^2 - 4^2

, which factors into

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

. Finally, factor the remaining difference of squares,

x^2 - 4

, into

(x - 2)(x + 2)

. The sum of squares,

x^2 + 4

, cannot be factored further. Combining these parts gives the complete factorization.

Adım Adım Çözüm

1

Factor out the greatest common factor from the polynomial.

2(x^4 - 16)

Both terms of

2x^4 - 32

are divisible by

2

, so we factor it out to simplify the remaining expression.

2

Factor the difference of squares inside the parentheses.

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

The expression

x^4 - 16

is a difference of squares because it can be written as

(x^2)^2 - 4^2

.

3

Factor the remaining difference of squares.

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

The binomial

x^2 - 4

is also a difference of squares (

x^2 - 2^2

), which factors into

(x - 2)(x + 2)

. The sum of squares

x^2 + 4

cannot be factored further using real numbers.

Anahtar Kavram

Factoring polynomials completely by extracting the greatest common factor and repeatedly applying the difference of squares formula.

Tahmini Süre:1m 0s

Soru 15Soru →

The square of $3$ less than a number $x$ is equal to $16$ . What is the greater of the two possible values for $x$ ?

5

11

7

19

25

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

The word problem translates to the equation

(x-3)^2 = 16

. Expanding the left side and subtracting 16 from both sides gives the standard quadratic equation

x^2 - 6x - 7 = 0

. This factors into

(x-7)(x+1) = 0

, yielding two solutions:

x = 7

and

x = -1

. The greater of these two values is 7.

Adım Adım Çözüm

1

Translate the verbal description into an algebraic equation.

(x-3)^2 = 16

'3 less than a number

x

' is written as

x - 3

, and its square is set equal to 16.

2

Expand the squared binomial and rearrange the terms into standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 6x - 7 = 0

Expanding

(x-3)^2

yields

x^2 - 6x + 9

. Subtracting 16 from both sides sets the equation to zero.

3

Factor the quadratic equation over the integers.

(x - 7)(x + 1) = 0

We find two integers that multiply to

-7

and add to

-6

, which are

-7

and

1

.

4

Solve for

x

by setting each factor to zero, then select the greater value.

x = 7

(since the solutions are

7

and

-1

)

Setting the factors to zero gives

x - 7 = 0 \implies x = 7

and

x + 1 = 0 \implies x = -1

. The larger of these two values is

7

.

Anahtar Kavram

Solving Quadratic Equations by Factoring

Tahmini Süre:1m 0s

Soru 16Soru →

Which of the following is a factor of the polynomial $8x^3 - 12x^2 - 18x + 27$ ?

2x - 9

4x - 9

4x - 3

2x + 3

2x + 9

Cevabı ve açıklamayı göster

Cevap: $2x + 3$

Cevap

The correct answer is

2x + 3

.

The correct answer is the factor

2x + 3

. By grouping the first two terms and the last two terms, we get

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

. Factoring out the common binomial yields

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

. The term

4x^2 - 9

is a difference of squares, which factors into

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

. Therefore, the fully factored expression is

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

, making the linear expression with a positive constant term the correct factor.

Adım Adım Çözüm

1

Group the four terms of the polynomial into two pairs.

(8x^3 - 12x^2) - (18x - 27)

Grouping terms allows us to find common binomial factors in a cubic polynomial.

2

Factor out the greatest common factor (GCF) from each group.

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

The GCF of

8x^3

and

12x^2

is

4x^2

, and the GCF of

-18x

and

27

is

-9

(factoring out the negative to match the binomials).

3

Factor out the common binomial factor

(2x - 3)

.

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

Both groups share the factor

(2x - 3)

, so it can be factored out.

4

Factor the quadratic difference of squares

(4x^2 - 9)

.

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

The term

4x^2 - 9

is a difference of squares of the form

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

, where

a = 2x

and

b = 3

.

Anahtar Kavram

Factoring a cubic polynomial completely by grouping and then applying the difference of squares formula.

Soru 17Soru →

The square of 3 less than twice a certain real number is equal to 3 less than 7 times that number. If $p$ and $q$ are the two distinct real solutions to this equation with $p > q$ , what is the value of $4p - 4q$ ?

9

11

13

14

1

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

13

Correctly expanding the squared expression leads to the quadratic equation

4x^2 - 19x + 12 = 0

. Factoring this equation yields the solutions

4

and

3/4

. Since the problem specifies

p > q

, we have

p = 4

and

q = 3/4

. Substituting these values into the expression

4p - 4q

gives

4(4) - 4(3/4) = 16 - 3 = 13

.

Adım Adım Çözüm

1

Translate the verbal description into an algebraic equation.

(2x - 3)^2 = 7x - 3

'Twice a number' is represented as

2x

, '3 less than twice the number' is

2x - 3

, and 'the square of' that is

(2x - 3)^2

. This is set equal to '3 less than 7 times that number', which is

7x - 3

.

2

Expand the binomial and write the equation in standard quadratic form

ax^2 + bx + c = 0

.

4x^2 - 19x + 12 = 0

Expanding the left side yields

4x^2 - 12x + 9 = 7x - 3

. Subtracting

7x

and adding

3

to both sides results in the standard form.

3

Factor the quadratic equation by grouping.

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

We need two numbers that multiply to

4 \times 12 = 48

and add to

-19

. These numbers are

-16

and

-3

. Rewriting the equation as

4x^2 - 16x - 3x + 12 = 0

allows us to factor out

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

, which simplifies to

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

.

4

Find the roots of the equation and evaluate the target expression

4p - 4q

.

p = 4

,

q = 3/4

, and the final value is

13

.

Setting each factor to zero gives

x = 3/4

and

x = 4

. Since

p > q

, we assign

p = 4

and

q = 3/4

. Evaluating the expression yields

4(4) - 4(3/4) = 16 - 3 = 13

.

Anahtar Kavram

Solving Quadratic Equations by Factoring

Tahmini Süre:2m 30s

Soru 18Soru →

When the expression $6x - 2x(x - 3y) - \left(2x - \frac{1}{2}y\right)^2 - \frac{3}{4}x(6x - 8)$ is simplified by combining like terms, what is the coefficient of the $x^2$ term?

-\frac{9}{2}

-5

-6

-\frac{17}{2}

-\frac{21}{2}

Cevabı ve açıklamayı göster

Cevap: $-\frac{21}{2}$

Cevap

-\frac{21}{2}

Expanding the entire expression yields the

x^2

terms

-2x^2

,

-4x^2

, and

-\frac{9}{2}x^2

. Summing these coefficients gives

-2 - 4 - \frac{9}{2} = -\frac{21}{2}

. Therefore, the coefficient of the

x^2

term is

-\frac{21}{2}

.

Adım Adım Çözüm

1

Expand the first parenthetical expression by distributing the term

-2x

.

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

. The expression becomes:

6x - 2x^2 + 6xy - \left(2x - \frac{1}{2}y\right)^2 - \frac{3}{4}x(6x - 8)

.

Distribution is required to eliminate parentheses before terms can be combined.

2

Expand the squared binomial

\left(2x - \frac{1}{2}y\right)^2

using the identity

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

, then distribute the negative sign.

\left(2x - \frac{1}{2}y\right)^2 = 4x^2 - 2xy + \frac{1}{4}y^2

. Distributing the negative gives

-4x^2 + 2xy - \frac{1}{4}y^2

.

This simplifies the second parenthetical term of the expression.

3

Expand the third parenthetical term by distributing

-\frac{3}{4}x

.

-\frac{3}{4}x(6x - 8) = -\frac{9}{2}x^2 + 6x

.

This simplifies the final parenthetical term of the expression.

4

Combine the coefficients of all the

x^2

terms.

The

x^2

terms are

-2x^2

,

-4x^2

, and

-\frac{9}{2}x^2

. Combining their coefficients gives:

-2 - 4 - \frac{9}{2} = -6 - \frac{9}{2} = -\frac{12}{2} - \frac{9}{2} = -\frac{21}{2}

.

Combining like terms simplifies the expression to find the final coefficient of

x^2

.

Anahtar Kavram

Simplifying algebraic expressions by distributing coefficients and combining like terms.

Tahmini Süre:1m 30s

Soru 19Soru →

An algebraic expression of the form $x^4 - 10x^2y^2 + 9y^4 - x^2 + 9y^2$ is defined for all real numbers $x$ and $y$ . When this expression is factored completely over the integers, which of the following is one of its factors?

x - 9y^2

x - y - 1

x - y

x + 3y

x^2 - y^2 + 1

Cevabı ve açıklamayı göster

Cevap: $x + 3y$

Cevap

The binomial

x + 3y

is a factor of the expression.

To factor the polynomial completely, we group the terms as

(x^4 - 10x^2y^2 + 9y^4) - (x^2 - 9y^2)

. The first trinomial is in quadratic form and factors to

(x^2 - 9y^2)(x^2 - y^2)

. The expression is then rewritten as

(x^2 - 9y^2)(x^2 - y^2) - (x^2 - 9y^2)

, which has a GCF of

(x^2 - 9y^2)

. Factoring out the GCF yields

(x^2 - 9y^2)(x^2 - y^2 - 1)

. Finally, the difference of squares

(x^2 - 9y^2)

factors into

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

. The fully factored expression is

(x - 3y)(x + 3y)(x^2 - y^2 - 1)

, which contains the factor

x + 3y

.

Adım Adım Çözüm

1

Group the terms of the polynomial into two parts.

(x^4 - 10x^2y^2 + 9y^4) - (x^2 - 9y^2)

Grouping terms allows us to find and extract common algebraic factors from distinct parts of the polynomial.

2

Factor the trinomial from the first group as a quadratic form in terms of

x^2

and

y^2

.

(x^2 - 9y^2)(x^2 - y^2)

The trinomial

x^4 - 10x^2y^2 + 9y^4

can be written as

(x^2)^2 - 10(x^2)(y^2) + 9(y^2)^2

, which factors as

(x^2 - 9y^2)(x^2 - y^2)

.

3

Substitute this factorization back into the grouped expression and factor out the greatest common factor.

(x^2 - 9y^2)(x^2 - y^2 - 1)

Both parts of the grouped expression share

(x^2 - 9y^2)

as a GCF, leaving

(x^2 - y^2 - 1)

when factored out.

4

Factor the difference of squares term completely over the integers.

(x - 3y)(x + 3y)(x^2 - y^2 - 1)

The term

(x^2 - 9y^2)

is a difference of squares of the form

a^2 - b^2

, which factors into

(a - b)(a + b)

.

Anahtar Kavram

Factoring high-degree polynomials using quadratic form substitution, grouping, and difference of squares.

Soru 20Soru →

The sum of $9$ and the product of a number and $6$ less than that number is equal to $3$ more than the number. What is the sum of all possible values of this number?

7

1

5

-2

-7

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

The correct answer is

7

. The verbal description translates to

x(x - 6) + 9 = x + 3

. Distributing the left side gives

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

. Subtracting

x

and

3

from both sides results in

x^2 - 7x + 6 = 0

. Factoring the quadratic expression gives

(x - 6)(x - 1) = 0

, which yields the solutions

x = 6

and

x = 1

. Summing these values gives

6 + 1 = 7

.

Adım Adım Çözüm

1

Translate the verbal statement into an algebraic equation.

x(x - 6) + 9 = x + 3

, where

x

represents the unknown number.

Establishing the relationship between the algebraic expressions defined by the problem.

2

Expand the left side of the equation and combine like terms to set the equation to zero.

x^2 - 6x + 9 = x + 3 \implies x^2 - 7x + 6 = 0

.

Quadratic equations must be set to zero before they can be solved by factoring.

3

Factor the quadratic equation over the integers.

(x - 6)(x - 1) = 0

.

Finding two numbers that multiply to

6

and add up to

-7

allows us to factor the trinomial.

4

Apply the zero-product property to find the individual roots.

x = 6

and

x = 1

.

If the product of two factors is zero, at least one of the factors must be zero.

5

Sum the possible values of the number.

6 + 1 = 7

.

The question asks for the sum of all possible values of the number.

Anahtar Kavram

Solving Quadratic Equations by Factoring

Elementary Algebra

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