Questions: Solve the equation by factoring. 8-8x=(3x+1)(x-1)

Transcript text: Solve the equation by factoring. \[ 8-8 x=(3 x+1)(x-1) \]

Solution

Solution Steps

To solve the equation by factoring, we first need to expand the right-hand side and then move all terms to one side of the equation to set it to zero. After that, we can factor the resulting quadratic equation and solve for \( x \).

Step 1: Expand the Right-Hand Side

First, we expand the right-hand side of the equation: \[ (3x + 1)(x - 1) = 3x^2 - 3x + x - 1 = 3x^2 - 2x - 1 \]

Step 2: Move All Terms to One Side

Next, we move all terms to one side of the equation to set it to zero: \[ 8 - 8x - (3x^2 - 2x - 1) = 0 \] Simplifying this, we get: \[ 8 - 8x - 3x^2 + 2x + 1 = 0 \] \[ -3x^2 - 6x + 9 = 0 \]

Step 3: Factor the Quadratic Equation

We factor the quadratic equation: \[ -3x^2 - 6x + 9 = 0 \] Factoring out \(-3\), we get: \[ -3(x^2 + 2x - 3) = 0 \] Solving \(x^2 + 2x - 3 = 0\), we factor it as: \[ (x + 3)(x - 1) = 0 \]

Step 4: Solve for \( x \)

Setting each factor to zero, we solve for \( x \): \[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \] \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]