(2x-7)(x-2)=(x+10)(x-5)

3 min read Jun 16, 2024

Solving the Equation: (2x-7)(x-2) = (x+10)(x-5)

This article will guide you through solving the equation (2x-7)(x-2) = (x+10)(x-5). We'll break down each step to make the process clear and understandable.

Expanding Both Sides

First, we need to expand both sides of the equation by using the distributive property (also known as FOIL).

Left Side: (2x-7)(x-2) = 2x(x-2) - 7(x-2) = 2x² - 4x - 7x + 14 = 2x² - 11x + 14
Right Side: (x+10)(x-5) = x(x-5) + 10(x-5) = x² - 5x + 10x - 50 = x² + 5x - 50

Now our equation looks like this: 2x² - 11x + 14 = x² + 5x - 50

Combining Like Terms

To simplify the equation further, we need to move all terms to one side.

Subtract x² from both sides: 2x² - 11x + 14 - x² = x² + 5x - 50 - x² This gives us: x² - 11x + 14 = 5x - 50
Subtract 5x from both sides: x² - 11x + 14 - 5x = 5x - 50 - 5x This gives us: x² - 16x + 14 = -50
Add 50 to both sides: x² - 16x + 14 + 50 = -50 + 50 This gives us: x² - 16x + 64 = 0

Factoring the Quadratic Equation

Now we have a quadratic equation in standard form: x² - 16x + 64 = 0. We can solve this by factoring.

Factor: Notice that the equation is a perfect square trinomial: (x - 8)² = 0
Take the square root of both sides: x - 8 = 0
Solve for x: x = 8

Solution

Therefore, the solution to the equation (2x-7)(x-2) = (x+10)(x-5) is x = 8.