(x-2)(5x+3)=(x-2)(3x-5)

3 min read Jun 17, 2024

Solving the Equation: (x-2)(5x+3) = (x-2)(3x-5)

This article will guide you through the steps to solve the equation: (x-2)(5x+3) = (x-2)(3x-5)

Understanding the Equation

The given equation involves two products of binomials. We can solve it by using the distributive property (also known as FOIL method) and then rearranging terms to get a quadratic equation.

Steps to Solve the Equation

Expand both sides of the equation:
- (x-2)(5x+3) = 5x² - 7x - 6
- (x-2)(3x-5) = 3x² - 11x + 10
Substitute the expanded terms back into the original equation:
- 5x² - 7x - 6 = 3x² - 11x + 10
Simplify by moving all terms to one side:
- 5x² - 3x² - 7x + 11x - 6 - 10 = 0
- 2x² + 4x - 16 = 0
Solve the quadratic equation:
- We can simplify the equation by dividing both sides by 2: x² + 2x - 8 = 0
- Now, we can factor the quadratic equation: (x+4)(x-2) = 0
- This gives us two possible solutions: x = -4 or x = 2

Verification

To ensure the solutions are correct, we can substitute them back into the original equation.

For x = -4:
- (-4-2)(5(-4)+3) = (-4-2)(3(-4)-5)
- (-6)(-17) = (-6)(-17)
- This verifies the solution x = -4
For x = 2:
- (2-2)(5(2)+3) = (2-2)(3(2)-5)
- (0)(13) = (0)(1)
- This verifies the solution x = 2

Conclusion

The equation (x-2)(5x+3) = (x-2)(3x-5) has two solutions: x = -4 and x = 2. We obtained these solutions by expanding the equation, rearranging terms, and solving the resulting quadratic equation. By substituting the solutions back into the original equation, we confirmed their validity.