Solving Quadratic Equations by Factoring Alıştırma Soruları — ACT

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

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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 2Soru →

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 3Soru →

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}

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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 4Soru →

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

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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 5Soru →

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

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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 6Soru →

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

Soru 7Soru →

Solve each quadratic equation by factoring, and match the equation to its correct solution set.

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Cevabı ve açıklamayı göster

Cevap

The equation

3x(4x + 5) = 5 - 2x

matches the solution set

\{-\frac{5}{3}, \frac{1}{4}\}

; the equation

x(12x + 1) = 35

matches the solution set

\{-\frac{7}{4}, \frac{5}{3}\}

; and the equation

(2x - 3)(3x + 1) = 5(1 - x)

matches the solution set

\{-1, \frac{4}{3}\}

.

Each of the quadratic equations is solved by first expanding any products, collecting all terms on the left-hand side to establish the standard form

ax^2 + bx + c = 0

, dividing by any common numerical factors, factoring the resulting quadratic expression into two linear binomials, and solving each linear equation for

x

. This correctly pairs the first equation with

\{-\frac{5}{3}, \frac{1}{4}\}

, the second equation with

\{-\frac{7}{4}, \frac{5}{3}\}

, and the third equation with

\{-1, \frac{4}{3}\}

.

Adım Adım Çözüm

1

Solve

3x(4x + 5) = 5 - 2x

.

12x^2 + 17x - 5 = 0 \Rightarrow (3x + 5)(4x - 1) = 0 \Rightarrow x = -\frac{5}{3}

or

x = \frac{1}{4}

.

Distribute the term on the left, rearrange the terms to set the equation to zero, find factors of

12 \times (-5) = -60

that sum to

17

(which are

20

and

-3

), factor by grouping, and apply the Zero Product Property.

2

Solve

x(12x + 1) = 35

.

12x^2 + x - 35 = 0 \Rightarrow (3x - 5)(4x + 7) = 0 \Rightarrow x = \frac{5}{3}

or

x = -\frac{7}{4}

.

Expand the left side, subtract

35

from both sides, find factors of

12 \times (-35) = -420

that sum to

1

(which are

21

and

-20

), factor by grouping, and solve for

x

.

3

Solve

(2x - 3)(3x + 1) = 5(1 - x)

.

6x^2 - 2x - 8 = 0 \Rightarrow 3x^2 - x - 4 = 0 \Rightarrow (3x - 4)(x + 1) = 0 \Rightarrow x = \frac{4}{3}

or

x = -1

.

Expand both sides, move all terms to the left, divide the quadratic equation by

2

to simplify, factor the trinomial, and solve for the roots.

Anahtar Kavram

Rearranging non-standard quadratic equations into the standard form

ax^2 + bx + c = 0

and solving them by factoring over the integers.

Soru 8Soru →

Match each quadratic equation with its correct solution set by solving the equation by factoring.

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Cevabı ve açıklamayı göster

Cevap

The equation

x(x - 1) = 12

matches the solution set

\{-3, 4\}

; the equation

2x^2 + 5x = 3

matches the solution set

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

; the equation

3x^2 + 8 = 10x

matches the solution set

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

; and the equation

2x^2 - 24 = 8x

matches the solution set

\{-2, 6\}

.

Each quadratic equation is correctly matched to its solutions by first rewriting the equation in standard form

ax^2 + bx + c = 0

, factoring the trinomial over the integers, and then applying the zero product property to find the roots.

Adım Adım Çözüm

1

Set each quadratic equation to standard form

ax^2 + bx + c = 0

by expanding terms and moving all terms to one side.

The equations become:
1)

x^2 - x - 12 = 0

2)

2x^2 + 5x - 3 = 0

3)

3x^2 - 10x + 8 = 0

4)

2x^2 - 8x - 24 = 0

Before a quadratic equation can be solved by factoring, it must be set equal to zero so that the zero product property can be applied.

2

Factor each quadratic expression completely over the integers.

The factored expressions are:
1)

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

2)

(2x - 1)(x + 3) = 0

3)

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

4)

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

Factoring rewrites the quadratic expressions as products of linear factors.

3

Apply the zero product property by setting each linear factor equal to zero and solving for

x

.

The solution sets are:
1)

x = 4

or

x = -3

, yielding

\{-3, 4\}

2)

x = \frac{1}{2}

or

x = -3

, yielding

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

3)

x = \frac{4}{3}

or

x = 2

, yielding

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

4)

x = 6

or

x = -2

, yielding

\{-2, 6\}

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

Anahtar Kavram

Solving Quadratic Equations by Factoring

Alternatif Yöntem

You can verify the solution sets by substituting the values of the roots back into the original equations to check if they yield a true statement, or by using the quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

as an alternative algebraic method.

Tahmini Süre:1m 30s

Soru 9Soru →

Match each quadratic equation with its correct set of real solutions.

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Cevabı ve açıklamayı göster

Cevap

The equation

x^2 - 5x + 6 = 0

matches with the solutions

x = 2

and

x = 3

. The equation

x^2 + 5x + 6 = 0

matches with the solutions

x = -3

and

x = -2

. The equation

x^2 - x - 6 = 0

matches with the solutions

x = -2

and

x = 3

.

Each equation is solved by factoring the quadratic trinomial into two binomials, then applying the zero product property to find the values of

x

that make each factor zero. Specifically,

x^2 - 5x + 6 = 0

factors into

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

, yielding solutions

x = 2

and

x = 3

. The equation

x^2 + 5x + 6 = 0

factors into

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

, yielding solutions

x = -2

and

x = -3

. Finally,

x^2 - x - 6 = 0

factors into

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

, yielding solutions

x = 3

and

x = -2

.

Adım Adım Çözüm

1

Factor the quadratic equation

x^2 - 5x + 6 = 0

.

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

Identify two integers that multiply to

6

and add up to

-5

. These integers are

-2

and

-3

.

2

Solve for

x

by setting each linear factor equal to zero:

x - 2 = 0

and

x - 3 = 0

.

x = 2

and

x = 3

Applying the zero product property means if the product of two numbers is zero, at least one of them must be zero.

3

Factor the quadratic equation

x^2 + 5x + 6 = 0

.

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

Identify two integers that multiply to

6

and add up to

5

. These integers are

2

and

3

.

4

Solve for

x

by setting each linear factor equal to zero:

x + 2 = 0

and

x + 3 = 0

.

x = -2

and

x = -3

Applying the zero product property gives the solutions as the negations of the terms inside the binomials.

5

Factor the quadratic equation

x^2 - x - 6 = 0

.

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

Identify two integers that multiply to

-6

and add up to

-1

. These integers are

-3

and

2

.

6

Solve for

x

by setting each linear factor equal to zero:

x - 3 = 0

and

x + 2 = 0

.

x = 3

and

x = -2

Setting the linear factors to zero yields the roots of the equation.

Anahtar Kavram

Solving Quadratic Equations by Factoring

Tahmini Süre:1m 30s

Soru 10Soru →

What is the sum of the solutions to the quadratic equation $x(x - 6) = 16$ ?

-6

10

6

12

38

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The sum of the solutions is 6.

Rearranging the equation to standard form gives

x^2 - 6x - 16 = 0

. Factoring the trinomial yields

(x - 8)(x + 2) = 0

. Solving for

x

by setting each factor to zero gives

x = 8

and

x = -2

. The sum of these solutions is

8 + (-2) = 6

.

Adım Adım Çözüm

1

Distribute the variable on the left side of the equation.

x^2 - 6x = 16

To solve a quadratic equation, we must first expand all products to identify the quadratic terms.

2

Subtract 16 from both sides of the equation to write it in standard form.

x^2 - 6x - 16 = 0

A quadratic equation must be set to zero before factoring.

3

Factor the quadratic trinomial.

(x - 8)(x + 2) = 0

We need to find two numbers that multiply to

-16

and add to

-6

. These numbers are

-8

and

2

.

4

Set each factor to zero and solve for

x

.

x = 8

and

x = -2

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

5

Add the two solutions to find their sum.

8 + (-2) = 6

The question asks for the sum of the solutions.

Anahtar Kavram

Solving quadratic equations by factoring after rearranging terms into standard form.

Soru 11Soru →

What is the sum of the distinct real solutions to the equation $(2x - 1)^2 = (x + 2)^2$ ?

\frac{8}{3}

0

\frac{4}{3}

8

3

Cevabı ve açıklamayı göster

Cevap: $\frac{8}{3}$

Cevap

The sum of the distinct real solutions is

\frac{8}{3}

.

To solve

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

, we rearrange the equation to

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

. Using the difference of squares identity, we factor this into

[(2x - 1) - (x + 2)][(2x - 1) + (x + 2)] = 0

, which simplifies to

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

. The solutions are

x = 3

and

x = -\frac{1}{3}

. Summing these gives

3 + (-\frac{1}{3}) = \frac{8}{3}

.

Adım Adım Çözüm

1

Rearrange the equation by moving all terms to one side to set it to zero.

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

To solve a quadratic equation by factoring, it must first be set equal to zero.

2

Factor the expression using the difference of squares formula,

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

, where

a = 2x - 1

and

b = x + 2

.

[(2x - 1) - (x + 2)][(2x - 1) + (x + 2)] = 0

Using the difference of squares allows us to factor the quadratic expression directly without fully expanding it.

3

Simplify the terms inside each set of brackets.

(2x - 1 - x - 2)(2x - 1 + x + 2) = 0 \Rightarrow (x - 3)(3x + 1) = 0

Simplifying the binomials reveals the two linear factors of the quadratic equation.

4

Set each linear factor to zero to find the distinct real solutions.

x - 3 = 0 \Rightarrow x = 3

and

3x + 1 = 0 \Rightarrow x = -\frac{1}{3}

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

5

Calculate the sum of the distinct real solutions.

3 + \left(-\frac{1}{3}\right) = \frac{9}{3} - \frac{1}{3} = \frac{8}{3}

The question asks for the sum of the solutions.

Anahtar Kavram

Solving quadratic equations by factoring, specifically utilizing the difference of squares method after rearranging terms.

Soru 12Soru →

A rectangular garden is surrounded by a uniform gravel path that is $1$ foot wide. The length of the garden is $3$ feet less than twice its width. If the total area of the garden and the path combined is $117$ square feet, what is the width of the garden, in feet?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The width of the garden is 7 feet.

By representing the garden's width as

w

, the length is

2w - 3

. The combined dimensions including the 1-foot uniform path on all sides are

w + 2

and

2w - 1

. Setting their product equal to the combined area of 117 square feet gives

(w+2)(2w-1) = 117

, which simplifies to the quadratic equation

2w^2 + 3w - 119 = 0

. Factoring this equation yields

(2w + 17)(w - 7) = 0

, giving the solutions

w = -8.5

and

w = 7

. Since width must be positive, the width of the garden is

7

feet.

Adım Adım Çözüm

1

Define variables for the garden's dimensions and the combined dimensions including the path.

Garden width =

w

, garden length =

2w - 3

. Combined width =

w + 2

, combined length =

2w - 1

.

The path surrounds the garden uniformly, adding

1

foot of width to each of the four sides (adding

2

feet total to both overall width and overall length).

2

Write the area equation for the combined area.

(w + 2)(2w - 1) = 117

The total area of the garden and path combined is given as

117

square feet.

3

Expand and rearrange the equation into standard quadratic form

aw^2 + bw + c = 0

.

2w^2 + 3w - 119 = 0

Expanding

(w + 2)(2w - 1)

gives

2w^2 + 3w - 2

. Subtracting

117

from both sides yields the standard form.

4

Factor the quadratic equation over the integers.

(2w + 17)(w - 7) = 0

We find two numbers that multiply to

2 \times (-119) = -238

and sum to

3

. These numbers are

17

and

-14

. Rewriting the middle term and factoring by grouping yields

(2w + 17)(w - 7) = 0

.

5

Solve for

w

and select the mathematically and physically valid solution.

w = 7

(discarding the negative root

w = -8.5

)

A physical measurement like width must be positive.

Anahtar Kavram

Solving quadratic word problems by setting up a quadratic equation and solving it by factoring.

Tahmini Süre:2m 0s

Soru 13Soru →

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 14Soru →

A certain real number $x$ satisfies the condition that the square of $3$ less than twice $x$ is equal to $8$ times the quantity $3$ minus $x$ . What is the ratio of the larger solution to the smaller solution of this equation?

-\frac{15}{4}

-\frac{5}{3}

-\frac{3}{5}

\frac{5}{3}

3

Cevabı ve açıklamayı göster

Cevap: $-\frac{5}{3}$

Cevap

The ratio of the larger solution to the smaller solution is

-\frac{5}{3}

.

The correct answer is

-\frac{5}{3}

. The verbal statement translates directly to

(2x - 3)^2 = 8(3 - x)

. Expanding both sides yields

4x^2 - 12x + 9 = 24 - 8x

. Rearranging into standard form gives

4x^2 - 4x - 15 = 0

. Factoring by grouping yields

(2x + 3)(2x - 5) = 0

, giving the solutions

x = -\frac{3}{2}

and

x = \frac{5}{2}

. The ratio of the larger root to the smaller root is

\frac{5/2}{-3/2} = -\frac{5}{3}

.

Adım Adım Çözüm

1

Translate the verbal description into an algebraic equation.

(2x - 3)^2 = 8(3 - x)

'Twice

x

' is

2x

, '3 less than twice

x

' is

2x - 3

, and its square is

(2x - 3)^2

. This is equal to 8 times the quantity

3 - x

.

2

Expand both sides of the equation.

4x^2 - 12x + 9 = 24 - 8x

Expanding the binomial

(2x - 3)^2

gives

4x^2 - 12x + 9

and distributing the right side gives

24 - 8x

.

3

Rearrange the equation to set it equal to zero.

4x^2 - 4x - 15 = 0

Add

8x

and subtract

24

from both sides to gather all terms on one side of the equation.

4

Factor the quadratic equation by grouping.

(2x + 3)(2x - 5) = 0

Find two numbers that multiply to

4 \times (-15) = -60

and add to

-4

. These numbers are

-10

and

6

. Rewrite the middle term as

-10x + 6x

and factor:

2x(2x - 5) + 3(2x - 5) = 0 \implies (2x + 3)(2x - 5) = 0

.

5

Solve for the roots of the equation.

x = -\frac{3}{2}

and

x = \frac{5}{2}

Set each factor equal to zero:

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

and

2x - 5 = 0 \implies x = \frac{5}{2}

.

6

Identify the larger and smaller solutions and compute their ratio.

-\frac{5}{3}

The larger solution is

\frac{5}{2}

and the smaller solution is

-\frac{3}{2}

. Their ratio is

\frac{5/2}{-3/2} = -\frac{5}{3}

.

Anahtar Kavram

Solving quadratic equations of the form

ax^2 + bx + c = 0

by factoring over the integers.

Soru 15Soru →

What is the sum of all real values of $x$ that satisfy the equation $(x - 3)^2 + x(x + 2) = 15$ ?

-2

-1

0

2

4

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The sum of all real values of

x

that satisfy the equation is

2

.

Expanding the equation yields

2x^2 - 4x + 9 = 15

. Setting this to zero gives

2x^2 - 4x - 6 = 0

. Dividing by the common factor of

2

simplifies this to

x^2 - 2x - 3 = 0

. Factoring the trinomial yields

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

, which gives the two solutions

x = 3

and

x = -1

. Summing these two solutions gives

3 + (-1) = 2

.

Adım Adım Çözüm

1

Expand both terms on the left side of the equation.

(x^2 - 6x + 9) + (x^2 + 2x) = 15

Applying the binomial squaring formula

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

to

(x - 3)^2

and distributing

x

to both terms in

x(x + 2)

allows us to simplify the equation.

2

Combine like terms and set the quadratic equation to zero.

2x^2 - 4x - 6 = 0

Grouping

x^2

terms,

x

terms, and constant terms on one side is necessary to format the quadratic equation as

ax^2 + bx + c = 0

before factoring.

3

Divide the entire equation by the common factor of

2

to simplify factoring.

x^2 - 2x - 3 = 0

Simplifying the quadratic equation makes it easier to find two binomial factors.

4

Factor the quadratic trinomial by finding two numbers that multiply to

-3

and add to

-2

.

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

Since

-3 \times 1 = -3

and

-3 + 1 = -2

, we can write the quadratic in factored form.

5

Set each factor to zero to solve for

x

.

x = 3

or

x = -1

Applying the zero product property determines the two values of

x

that satisfy the original equation.

6

Calculate the sum of the two solutions.

3 + (-1) = 2

The question asks for the sum of all real values of

x

that satisfy the equation.

Anahtar Kavram

Solving quadratic equations by rearranging terms, factoring trinomials, and applying the zero product property.

Soru 16Soru →

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 17Soru →

If $x(x - 6) = 7$ , what is the positive difference between the two solutions to this equation?

6

8

10

14

0

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The positive difference between the two solutions is

8

.

First, expand the left side of the equation to get

x^2 - 6x = 7

. Next, subtract

7

from both sides to write the equation in standard form:

x^2 - 6x - 7 = 0

. Factoring this quadratic expression gives

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

. Setting each factor to zero yields the solutions

x = 7

and

x = -1

. The positive difference between these two solutions is

7 - (-1) = 8

.

Adım Adım Çözüm

1

Distribute the

x

on the left side of the equation.

x^2 - 6x = 7

Multiplying

x

by

(x - 6)

expands the left side to prepare for standard form.

2

Subtract

7

from both sides to set the quadratic equation to zero.

x^2 - 6x - 7 = 0

Setting the quadratic equation to standard form

ax^2 + bx + c = 0

is necessary before factoring.

3

Factor the quadratic trinomial.

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

We need two numbers that multiply to

-7

and add to

-6

, which are

-7

and

1

.

4

Set each factor to zero to find the solutions.

x = 7

and

x = -1

The zero product property states that if a product is zero, at least one factor must be zero.

5

Calculate the positive difference between the two solutions.

7 - (-1) = 8

The positive difference is found by subtracting the smaller solution from the larger solution.

Anahtar Kavram

Solving Quadratic Equations by Factoring

Tahmini Süre:1m 0s

Soru 18Soru →

What is the positive difference between the two real solutions to the equation $(2x - 1)^2 = x(3x - 5) + 7$ ?

1

3

5

6

7

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The positive difference between the two real solutions is 5.

The correct answer is 5. By expanding the equation, we get

4x^2 - 4x + 1 = 3x^2 - 5x + 7

. Rearranging the terms to set the equation to zero yields

x^2 + x - 6 = 0

. Factoring this equation gives

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

, which has the solutions

x = -3

and

x = 2

. The positive difference between these two solutions is

|2 - (-3)| = 5

.

Adım Adım Çözüm

1

Expand both sides of the equation.

4x^2 - 4x + 1 = 3x^2 - 5x + 7

Expanding the squared term on the left side and distributing the

x

on the right side allows us to write the equation in polynomial form.

2

Rearrange the equation by moving all terms to the left side to set the right side to zero.

x^2 + x - 6 = 0

Subtracting

3x^2

, adding

5x

, and subtracting

7

from both sides simplifies the equation into standard quadratic form,

ax^2 + bx + c = 0

.

3

Factor the quadratic trinomial.

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

Finding two integers that multiply to

-6

and add to

1

gives

3

and

-2

, allowing us to factor the equation over the integers.

4

Solve for

x

by setting each linear factor to zero.

x = -3

and

x = 2

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

5

Calculate the positive difference between the two solutions.

|2 - (-3)| = 5

Subtracting the smaller solution from the larger solution gives the positive distance between them on the number line.

Anahtar Kavram

Solving quadratic equations by expanding, rearranging into standard form, and factoring over the integers.

Tahmini Süre:2m 0s

Soru 19Soru →

What is the positive difference between the two real solutions to the equation $(x - 1)^2 = 5x - 5$ ?

1

5

7

6

4

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The positive difference between the two real solutions is 5.

Expanding the left side of

(x - 1)^2 = 5x - 5

yields

x^2 - 2x + 1 = 5x - 5

. Moving all terms to the left side by subtracting

5x

and adding

5

gives the standard quadratic equation

x^2 - 7x + 6 = 0

. Factoring this expression gives

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

, which yields the solutions

1

and

6

. The positive difference between these solutions is

6 - 1 = 5

.

Adım Adım Çözüm

1

Expand the squared binomial on the left side of the equation.

x^2 - 2x + 1 = 5x - 5

Before factoring a quadratic equation, all terms must be expanded and moved to one side to set the equation equal to zero.

2

Subtract

5x

and add

5

to both sides to rearrange the equation into standard quadratic form.

x^2 - 7x + 6 = 0

This sets the equation equal to zero, which is a prerequisite for using the zero product property.

3

Factor the quadratic expression by finding two numbers that multiply to

6

and add to

-7

.

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

Factoring allows us to split the quadratic equation into two linear equations.

4

Set each factor to zero to solve for

x

.

x = 1

and

x = 6

By the zero product property, if the product of two factors is zero, at least one of the factors must be zero.

5

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

6 - 1 = 5

The question asks for the positive difference between the two solutions.

Anahtar Kavram

Solving quadratic equations by expanding, rearranging into standard form, and factoring over the integers.

Soru 20Soru →

For each of the given quadratic equations, solve for $x$ by factoring. Match each quadratic equation on the left to its correct solution set on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The equation

2x^2 + 5x = 3

matches the solution set

\left\{-3, \frac{1}{2}\right\}

; the equation

3x^2 - 10x + 8 = 0

matches the solution set

\left\{\frac{4}{3}, 2\right\}

; and the equation

x(x + 2) = 15

matches the solution set

\left\{-5, 3\right\}

.

Each equation matches its corresponding solution set through distributing terms if necessary, rewriting the equation in standard quadratic form

ax^2 + bx + c = 0

, factoring the quadratic trinomial over the integers, and then using the zero product property to solve for

x

.

Adım Adım Çözüm

1

For the equation

2x^2 + 5x = 3

, rewrite in standard form

ax^2 + bx + c = 0

by subtracting 3 from both sides to get

2x^2 + 5x - 3 = 0

.

The equation is rewritten as

2x^2 + 5x - 3 = 0

.

Before factoring a quadratic equation, all terms must be moved to one side so the other side is equal to zero.

2

Factor the trinomial

2x^2 + 5x - 3

by grouping. Find two integers that multiply to

2 \times (-3) = -6

and add to

5

. These integers are

6

and

-1

. Rewrite the middle term and factor by grouping:

2x^2 + 6x - x - 3 = 2x(x + 3) - 1(x + 3) = (2x - 1)(x + 3) = 0

.

The equation becomes

(2x - 1)(x + 3) = 0

.

Factoring allows us to apply the zero product property to find the solutions.

3

Set each factor of

(2x - 1)(x + 3) = 0

to zero and solve for

x

:

2x - 1 = 0 \Rightarrow x = \frac{1}{2}

and

x + 3 = 0 \Rightarrow x = -3

.

The solutions are

x = \frac{1}{2}

and

x = -3

, forming the solution set

\left\{-3, \frac{1}{2}\right\}

.

By the zero product property, if the product of two factors is zero, at least one of the factors must be zero.

4

For the equation

3x^2 - 10x + 8 = 0

, factor the trinomial by finding two integers that multiply to

3 \times 8 = 24

and add to

-10

. These integers are

-6

and

-4

. Rewrite the middle term and factor by grouping:

3x^2 - 6x - 4x + 8 = 3x(x - 2) - 4(x - 2) = (3x - 4)(x - 2) = 0

.

The equation becomes

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

.

The equation is already in standard form, so we can directly proceed with factoring.

5

Set each factor of

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

to zero and solve for

x

:

3x - 4 = 0 \Rightarrow x = \frac{4}{3}

and

x - 2 = 0 \Rightarrow x = 2

.

The solutions are

x = \frac{4}{3}

and

x = 2

, forming the solution set

\left\{\frac{4}{3}, 2\right\}

.

Solving each linear factor yields the roots of the quadratic equation.

6

For the equation

x(x + 2) = 15

, first distribute

x

to get

x^2 + 2x = 15

, then subtract 15 from both sides to write in standard form:

x^2 + 2x - 15 = 0

.

The equation is rewritten as

x^2 + 2x - 15 = 0

.

Distributing and moving terms sets the quadratic to zero, which is necessary for factoring.

7

Factor the quadratic

x^2 + 2x - 15 = 0

by finding two integers that multiply to

-15

and add to

2

. These integers are

5

and

-3

, yielding

(x + 5)(x - 3) = 0

.

The equation becomes

(x + 5)(x - 3) = 0

.

Factoring a quadratic trinomial with a leading coefficient of 1 involves finding numbers that sum to the linear coefficient and multiply to the constant term.

8

Set each factor of

(x + 5)(x - 3) = 0

to zero and solve for

x

:

x + 5 = 0 \Rightarrow x = -5

and

x - 3 = 0 \Rightarrow x = 3

.

The solutions are

x = -5

and

x = 3

, forming the solution set

\left\{-5, 3\right\}

.

Solving the resulting linear equations gives the roots of the original quadratic equation.

Anahtar Kavram

Solving Quadratic Equations by Factoring

Alternatif Yöntem

Instead of factoring, the solutions to these quadratic equations can be verified by substituting the values in the solution sets back into the original equations, or by applying the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

after rewriting them in standard form.

Tahmini Süre:2m 0s

Solving Quadratic Equations by Factoring

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