(2x+9)(x-5)=108

3 min read Jun 16, 2024

Solving the Equation (2x + 9)(x - 5) = 108

This equation represents a quadratic equation in disguise. Let's break down the steps to solve it:

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

(2x + 9)(x - 5) = 2x² - 10x + 9x - 45 = 2x² - x - 45

Now our equation looks like this:

2x² - x - 45 = 108

To solve for x, we need to bring all terms to one side of the equation and set it equal to zero.

Subtract 108 from both sides:

2x² - x - 153 = 0

Now we have a standard quadratic equation in the form ax² + bx + c = 0. We can solve for x using several methods:

Factoring: Try to factor the quadratic expression. In this case, factoring might be tricky, so we'll explore other methods.
Quadratic Formula: The quadratic formula is a reliable way to find the solutions (roots) of any quadratic equation. The formula is: x = (-b ± √(b² - 4ac)) / 2a

In our equation, a = 2, b = -1, and c = -153. Substituting these values into the formula gives us:

x = (1 ± √((-1)² - 4 * 2 * -153)) / (2 * 2) x = (1 ± √(1225)) / 4 x = (1 ± 35) / 4

Therefore, we have two possible solutions:

x₁ = (1 + 35) / 4 = 9 x₂ = (1 - 35) / 4 = -8.5

To ensure our solutions are correct, we can plug them back into the original equation:

For x = 9: (2 * 9 + 9)(9 - 5) = 27 * 4 = 108 (This solution checks out)
For x = -8.5: (2 * -8.5 + 9)(-8.5 - 5) = -8 * -13.5 = 108 (This solution also checks out)

Therefore, the solutions to the equation (2x + 9)(x - 5) = 108 are x = 9 and x = -8.5.