Question Solve the equation $8-8x=(3x+1)(x-1)$ by factoring. The solution set is $x=1, -\frac{1}{3}$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the equation $8-8x = (3x+1)(x-1)$ .
2. We need to solve for $x$ by factoring.

STEP 2

First, expand the right-hand side of the equation to simplify it.
$(3x+1)(x-1) = 3x(x-1) + 1(x-1)$

STEP 3

Distribute the terms on the right-hand side.
$3x(x-1) + 1(x-1) = 3x^2 - 3x + x - 1$

STEP 4

Combine like terms on the right-hand side.
$3x^2 - 3x + x - 1 = 3x^2 - 2x - 1$

STEP 5

Rewrite the original equation with the expanded right-hand side.
$8 - 8x = 3x^2 - 2x - 1$

STEP 6

Move all terms to one side to set the equation to zero, which is necessary for factoring.
$3x^2 - 2x - 1 - 8 + 8x = 0$

STEP 7

Combine like terms.
$3x^2 + 6x - 9 = 0$

STEP 8

Now, we need to factor the quadratic equation $3x^2 + 6x - 9$ .

STEP 9

Look for two numbers that multiply to $3 \times -9 = -27$ and add up to $6$ . These numbers are $9$ and $-3$ .

STEP 10

Rewrite the middle term $6x$ using the numbers found in STEP_9.
$3x^2 + 9x - 3x - 9 = 0$

STEP 11

Factor by grouping. Group the terms into two pairs.
$(3x^2 + 9x) - (3x + 9) = 0$

STEP 12

Factor out the common factor from each group.
$3x(x + 3) - 3(x + 3) = 0$

STEP 13

Factor out the common binomial factor $(x + 3)$ .
$(3x - 3)(x + 3) = 0$

STEP 14

Set each factor equal to zero to find the solutions for $x$ .
$3x - 3 = 0 \quad \text{or} \quad x + 3 = 0$

STEP 15

Solve the first equation $3x - 3 = 0$ .
$3x = 3$

STEP 16

Divide both sides by $3$ to isolate $x$ .
$x = \frac{3}{3}$

STEP 17

Calculate the value of $x$ .
$x = 1$

STEP 18

Solve the second equation $x + 3 = 0$ .
$x = -3$

STEP 19

Combine the solutions to form the solution set.
The solution set is $\{1, -3\}$ .

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