Solved on Feb 12, 2024

Solve the equation $32-32x = (6x+4)(x-1)$ by factoring. The solution set is $\{-4, 1\}$ .

STEP 1

Assumptions
1. We are given the equation $32-32x=(6x+4)(x-1)$ .
2. We need to solve for $x$ by factoring.
3. We will use the zero-product property, which states that if $ab=0$ , then either $a=0$ or $b=0$ or both.

STEP 2

First, we will expand the right side of the equation to simplify it.
$(6x+4)(x-1) = 6x(x-1) + 4(x-1)$

STEP 3

Now, distribute the terms on the right side.
$6x(x-1) + 4(x-1) = 6x^2 - 6x + 4x - 4$

STEP 4

Combine like terms on the right side.
$6x^2 - 6x + 4x - 4 = 6x^2 - 2x - 4$

STEP 5

Rewrite the original equation with the expanded right side.
$32-32x = 6x^2 - 2x - 4$

STEP 6

To solve the equation by factoring, we need to set the equation to zero. To do this, we will subtract $32$ and add $32x$ to both sides of the equation.
$0 = 6x^2 - 2x - 4 + 32x - 32$

STEP 7

Combine like terms on the right side.
$0 = 6x^2 + 30x - 36$

STEP 8

Now, we will attempt to factor the quadratic equation. We are looking for two numbers that multiply to $6 \times -36 = -216$ and add to $30$ .

STEP 9

The two numbers that satisfy these conditions are $36$ and $-6$ , since $36 \times -6 = -216$ and $36 + (-6) = 30$ .

STEP 10

Rewrite the middle term $30x$ using the numbers $36$ and $-6$ .
$0 = 6x^2 + 36x - 6x - 36$

STEP 11

Factor by grouping. Group the first two terms and the last two terms.
$0 = (6x^2 + 36x) - (6x + 36)$

STEP 12

Factor out the common factor from each group.
$0 = 6x(x + 6) - 6(x + 6)$

STEP 13

Now, factor out the common binomial factor $(x + 6)$ .
$0 = (6x - 6)(x + 6)$

STEP 14

We can further factor out a $6$ from the first term.
$0 = 6(x - 1)(x + 6)$

STEP 15

Now, apply the zero-product property. Set each factor equal to zero.
$6 = 0, \quad x - 1 = 0, \quad x + 6 = 0$

STEP 16

Since $6$ is not equal to $0$ , we ignore the first equation. Solve the other two equations for $x$ .
$x - 1 = 0 \quad \Rightarrow \quad x = 1$ $x + 6 = 0 \quad \Rightarrow \quad x = -6$

STEP 17

The solution set for the equation is $\{1, -6\}$ .
The solution set is $\{1, -6\}$ .

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Solve the equation $32-32x = (6x+4)(x-1)$ by factoring. The solution set is $\{-4, 1\}$ .

STEP 1

Assumptions
1. We are given the equation $32-32x=(6x+4)(x-1)$ .
2. We need to solve for $x$ by factoring.
3. We will use the zero-product property, which states that if $ab=0$ , then either $a=0$ or $b=0$ or both.

STEP 2

First, we will expand the right side of the equation to simplify it.
$(6x+4)(x-1) = 6x(x-1) + 4(x-1)$

STEP 3

Now, distribute the terms on the right side.
$6x(x-1) + 4(x-1) = 6x^2 - 6x + 4x - 4$

STEP 4

Combine like terms on the right side.
$6x^2 - 6x + 4x - 4 = 6x^2 - 2x - 4$

STEP 5

Rewrite the original equation with the expanded right side.
$32-32x = 6x^2 - 2x - 4$

STEP 6

To solve the equation by factoring, we need to set the equation to zero. To do this, we will subtract $32$ and add $32x$ to both sides of the equation.
$0 = 6x^2 - 2x - 4 + 32x - 32$

STEP 7

Combine like terms on the right side.
$0 = 6x^2 + 30x - 36$

STEP 8

Now, we will attempt to factor the quadratic equation. We are looking for two numbers that multiply to $6 \times -36 = -216$ and add to $30$ .

STEP 9

The two numbers that satisfy these conditions are $36$ and $-6$ , since $36 \times -6 = -216$ and $36 + (-6) = 30$ .

STEP 10

Rewrite the middle term $30x$ using the numbers $36$ and $-6$ .
$0 = 6x^2 + 36x - 6x - 36$

STEP 11

Factor by grouping. Group the first two terms and the last two terms.
$0 = (6x^2 + 36x) - (6x + 36)$

STEP 12

Factor out the common factor from each group.
$0 = 6x(x + 6) - 6(x + 6)$

STEP 13

Now, factor out the common binomial factor $(x + 6)$ .
$0 = (6x - 6)(x + 6)$

STEP 14

We can further factor out a $6$ from the first term.
$0 = 6(x - 1)(x + 6)$

STEP 15

Now, apply the zero-product property. Set each factor equal to zero.
$6 = 0, \quad x - 1 = 0, \quad x + 6 = 0$

STEP 16

Since $6$ is not equal to $0$ , we ignore the first equation. Solve the other two equations for $x$ .
$x - 1 = 0 \quad \Rightarrow \quad x = 1$ $x + 6 = 0 \quad \Rightarrow \quad x = -6$

STEP 17

The solution set for the equation is $\{1, -6\}$ .
The solution set is $\{1, -6\}$ .

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Solve the equation 32−32x=(6x+4)(x−1)32-32x = (6x+4)(x-1)32−32x=(6x+4)(x−1) by factoring. The solution set is {−4,1}\{-4, 1\}{−4,1}.

STEP 1

Assumptions1. We are given the equation 32−32x=(6x+4)(x−1)32-32x=(6x+4)(x-1)32−32x=(6x+4)(x−1).2. We need to solve for xxx by factoring.3. We will use the zero-product property, which states that if ab=0ab=0ab=0, then either a=0a=0a=0 or b=0b=0b=0 or both.

STEP 2

First, we will expand the right side of the equation to simplify it.(6x+4)(x−1)=6x(x−1)+4(x−1)(6x+4)(x-1) = 6x(x-1) + 4(x-1)(6x+4)(x−1)=6x(x−1)+4(x−1)

STEP 3

Now, distribute the terms on the right side.6x(x−1)+4(x−1)=6x2−6x+4x−46x(x-1) + 4(x-1) = 6x^2 - 6x + 4x - 46x(x−1)+4(x−1)=6x2−6x+4x−4

STEP 4

Combine like terms on the right side.6x2−6x+4x−4=6x2−2x−46x^2 - 6x + 4x - 4 = 6x^2 - 2x - 46x2−6x+4x−4=6x2−2x−4

STEP 5

Rewrite the original equation with the expanded right side.32−32x=6x2−2x−432-32x = 6x^2 - 2x - 432−32x=6x2−2x−4

STEP 6

To solve the equation by factoring, we need to set the equation to zero. To do this, we will subtract 323232 and add 32x32x32x to both sides of the equation.0=6x2−2x−4+32x−320 = 6x^2 - 2x - 4 + 32x - 320=6x2−2x−4+32x−32

STEP 7

Combine like terms on the right side.0=6x2+30x−360 = 6x^2 + 30x - 360=6x2+30x−36

STEP 8

Now, we will attempt to factor the quadratic equation. We are looking for two numbers that multiply to 6×−36=−2166 \times -36 = -2166×−36=−216 and add to 303030.

STEP 9

The two numbers that satisfy these conditions are 363636 and −6-6−6, since 36×−6=−21636 \times -6 = -21636×−6=−216 and 36+(−6)=3036 + (-6) = 3036+(−6)=30.

STEP 10

Rewrite the middle term 30x30x30x using the numbers 363636 and −6-6−6.0=6x2+36x−6x−360 = 6x^2 + 36x - 6x - 360=6x2+36x−6x−36

STEP 11

Factor by grouping. Group the first two terms and the last two terms.0=(6x2+36x)−(6x+36)0 = (6x^2 + 36x) - (6x + 36)0=(6x2+36x)−(6x+36)

STEP 12

Factor out the common factor from each group.0=6x(x+6)−6(x+6)0 = 6x(x + 6) - 6(x + 6)0=6x(x+6)−6(x+6)

STEP 13

Now, factor out the common binomial factor (x+6)(x + 6)(x+6).0=(6x−6)(x+6)0 = (6x - 6)(x + 6)0=(6x−6)(x+6)

STEP 14

We can further factor out a 666 from the first term.0=6(x−1)(x+6)0 = 6(x - 1)(x + 6)0=6(x−1)(x+6)

STEP 15

Now, apply the zero-product property. Set each factor equal to zero.6=0,x−1=0,x+6=06 = 0, \quad x - 1 = 0, \quad x + 6 = 06=0,x−1=0,x+6=0

STEP 16

Since 666 is not equal to 000, we ignore the first equation. Solve the other two equations for xxx.x−1=0⇒x=1x - 1 = 0 \quad \Rightarrow \quad x = 1x−1=0⇒x=1 x+6=0⇒x=−6x + 6 = 0 \quad \Rightarrow \quad x = -6x+6=0⇒x=−6

STEP 17

The solution set for the equation is {1,−6}\{1, -6\}{1,−6}.The solution set is {1,−6}\{1, -6\}{1,−6}.

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Solve the equation $32-32x = (6x+4)(x-1)$ by factoring. The solution set is $\{-4, 1\}$ .

Assumptions
1. We are given the equation $32-32x=(6x+4)(x-1)$ .
2. We need to solve for $x$ by factoring.
3. We will use the zero-product property, which states that if $ab=0$ , then either $a=0$ or $b=0$ or both.

First, we will expand the right side of the equation to simplify it.
$(6x+4)(x-1) = 6x(x-1) + 4(x-1)$

Now, distribute the terms on the right side.
$6x(x-1) + 4(x-1) = 6x^2 - 6x + 4x - 4$

Combine like terms on the right side.
$6x^2 - 6x + 4x - 4 = 6x^2 - 2x - 4$

Rewrite the original equation with the expanded right side.
$32-32x = 6x^2 - 2x - 4$

To solve the equation by factoring, we need to set the equation to zero. To do this, we will subtract $32$ and add $32x$ to both sides of the equation.
$0 = 6x^2 - 2x - 4 + 32x - 32$

Combine like terms on the right side.
$0 = 6x^2 + 30x - 36$

Now, we will attempt to factor the quadratic equation. We are looking for two numbers that multiply to $6 \times -36 = -216$ and add to $30$ .

The two numbers that satisfy these conditions are $36$ and $-6$ , since $36 \times -6 = -216$ and $36 + (-6) = 30$ .

Rewrite the middle term $30x$ using the numbers $36$ and $-6$ .
$0 = 6x^2 + 36x - 6x - 36$

Factor by grouping. Group the first two terms and the last two terms.
$0 = (6x^2 + 36x) - (6x + 36)$

Factor out the common factor from each group.
$0 = 6x(x + 6) - 6(x + 6)$

Now, factor out the common binomial factor $(x + 6)$ .
$0 = (6x - 6)(x + 6)$

We can further factor out a $6$ from the first term.
$0 = 6(x - 1)(x + 6)$

Now, apply the zero-product property. Set each factor equal to zero.
$6 = 0, \quad x - 1 = 0, \quad x + 6 = 0$

Since $6$ is not equal to $0$ , we ignore the first equation. Solve the other two equations for $x$ .
$x - 1 = 0 \quad \Rightarrow \quad x = 1$ $x + 6 = 0 \quad \Rightarrow \quad x = -6$

The solution set for the equation is $\{1, -6\}$ .
The solution set is $\{1, -6\}$ .