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

3 min read Jun 17, 2024

Solving the Equation (x-1)(x+3) = 12

This article will guide you through the steps of solving the equation (x-1)(x+3) = 12.

First, we need to expand the left side of the equation by using the distributive property (or FOIL method):

(x-1)(x+3) = x² + 3x - x - 3 = x² + 2x - 3

Now the equation becomes:

x² + 2x - 3 = 12

To solve for x, we need to set the equation equal to zero. Subtract 12 from both sides:

x² + 2x - 3 - 12 = 0

This simplifies to:

x² + 2x - 15 = 0

We have a quadratic equation now. We can solve this equation by factoring, completing the square, or using the quadratic formula.

Factoring:

Find two numbers that add up to 2 (the coefficient of the x term) and multiply to -15 (the constant term).
These numbers are 5 and -3.
We can rewrite the equation as: (x+5)(x-3) = 0
To satisfy the equation, either (x+5) = 0 or (x-3) = 0.
Therefore, the solutions are x = -5 and x = 3.

Quadratic Formula:

The quadratic formula solves for x in any equation of the form ax² + bx + c = 0:

x = (-b ± √(b² - 4ac)) / 2a

In our case, a = 1, b = 2, and c = -15.

Plugging these values into the formula, we get:

x = (-2 ± √(2² - 4 * 1 * -15)) / (2 * 1)

x = (-2 ± √(64)) / 2

x = (-2 ± 8) / 2

This gives us two solutions:

x = (-2 + 8) / 2 = 3

x = (-2 - 8) / 2 = -5

Therefore, the solutions to the equation (x-1)(x+3) = 12 are x = -5 and x = 3.