(x-1)(2x-1)=(x+8)(x-1)

3 min read Jun 17, 2024

Solving the Equation (x-1)(2x-1) = (x+8)(x-1)

This equation involves a quadratic expression and presents a straightforward opportunity to practice algebraic manipulation and problem-solving techniques. Let's break down the steps to find the solution(s) for x.

1. Expand Both Sides

Begin by expanding both sides of the equation using the distributive property (also known as FOIL).

Left side: (x-1)(2x-1) = 2x² - x - 2x + 1 = 2x² - 3x + 1
Right side: (x+8)(x-1) = x² - x + 8x - 8 = x² + 7x - 8

The equation now becomes: 2x² - 3x + 1 = x² + 7x - 8

2. Simplify by Combining Like Terms

Move all terms to one side of the equation to set it equal to zero.

Subtract x² from both sides: x² - 3x + 1 = 7x - 8
Subtract 7x from both sides: x² - 10x + 1 = -8
Add 8 to both sides: x² - 10x + 9 = 0

3. Factor the Quadratic Expression

Now we have a quadratic equation in standard form (ax² + bx + c = 0). Factor the expression on the left-hand side.

Find two numbers that add up to -10 and multiply to 9: These numbers are -1 and -9.
Factor the expression: (x - 1)(x - 9) = 0

4. Solve for x

For the product of two factors to equal zero, at least one of them must be zero. Therefore, we have two possible solutions:

x - 1 = 0: This gives us x = 1
x - 9 = 0: This gives us x = 9

Conclusion

The solutions to the equation (x-1)(2x-1) = (x+8)(x-1) are x = 1 and x = 9.