(2x-1)2-(x+1)2=9

2 min read Jun 16, 2024

Solving the Equation: (2x-1)2 - (x+1)2 = 9

This article will guide you through the process of solving the equation (2x-1)2 - (x+1)2 = 9.

Understanding the Problem

The equation involves squares of binomial expressions and a constant. To solve it, we will utilize the algebraic concept of difference of squares.

Applying the Difference of Squares

The difference of squares states that: a² - b² = (a + b)(a - b)

We can apply this to our equation:

(2x-1)² - (x+1)² = (2x-1 + x+1)(2x-1 - x-1) = 9

Simplifying the equation further:

(3x)(x-2) = 9
3x² - 6x - 9 = 0

Solving the Quadratic Equation

Now we have a quadratic equation in the form ax² + bx + c = 0. To solve it, we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

In our case, a = 3, b = -6, and c = -9. Substituting these values into the formula:

x = (6 ± √((-6)² - 4 * 3 * -9)) / (2 * 3)
x = (6 ± √(144)) / 6
x = (6 ± 12) / 6

This gives us two solutions:

x1 = (6 + 12) / 6 = 3
x2 = (6 - 12) / 6 = -1

Conclusion

Therefore, the solutions to the equation (2x-1)2 - (x+1)2 = 9 are x = 3 and x = -1.