(x-5)(x+7)=12

2 min read Jun 17, 2024

Solving the Equation (x-5)(x+7) = 12

This article will guide you through solving the quadratic equation (x-5)(x+7) = 12. We'll cover the steps involved and provide explanations along the way.

1. Expanding the Equation

First, we need to expand the left side of the equation by multiplying the binomials:

(x-5)(x+7) = x² + 2x - 35

Now, we have: x² + 2x - 35 = 12

2. Rearranging the Equation

Next, we'll move the constant term from the right side to the left, making the equation equal to zero:

x² + 2x - 35 - 12 = 0

This simplifies to: x² + 2x - 47 = 0

3. Solving for x

Now we have a standard quadratic equation in the form ax² + bx + c = 0. We can solve this using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

In our equation, a = 1, b = 2, and c = -47. Substituting these values into the formula, we get:

x = (-2 ± √(2² - 4 * 1 * -47)) / (2 * 1)

Simplifying further:

x = (-2 ± √(192)) / 2

x = (-2 ± 8√3) / 2

Finally, we can simplify by dividing both numerator and denominator by 2:

x = -1 ± 4√3

Conclusion

Therefore, the solutions to the equation (x-5)(x+7) = 12 are x = -1 + 4√3 and x = -1 - 4√3.