(x-5)(x+5)-15x+75=0

2 min read Jun 17, 2024

Solving the Quadratic Equation: (x-5)(x+5) - 15x + 75 = 0

This article will guide you through solving the quadratic equation (x-5)(x+5) - 15x + 75 = 0. We'll break down the steps and explain the concepts involved.

1. Expanding the Equation

First, we need to simplify the equation by expanding the product of the binomials:

(x-5)(x+5) = x² - 25

Substituting this back into the original equation:

x² - 25 - 15x + 75 = 0

2. Combining Like Terms

Next, combine the constant terms:

x² - 15x + 50 = 0

3. Factoring the Quadratic Equation

Now, we factor the quadratic expression. We need to find two numbers that add up to -15 (the coefficient of the x term) and multiply to 50 (the constant term). These numbers are -10 and -5.

Therefore, we can factor the equation as:

(x - 10)(x - 5) = 0

4. Solving for x

To find the solutions, we set each factor equal to zero:

x - 10 = 0 or x - 5 = 0

Solving for x in each case:

x = 10 or x = 5

5. The Solutions

Therefore, the solutions to the quadratic equation (x-5)(x+5) - 15x + 75 = 0 are:

x = 10 and x = 5

This means that the equation is satisfied when x is equal to either 10 or 5.