%5E2%3D4x.jpeg "(2x-3)^2=4x")
Solving the Equation (2x-3)^2 = 4x
This article will guide you through solving the equation (2x-3)^2 = 4x. We will use algebraic manipulations to find the solutions for x.
Expanding the Equation
First, we need to expand the left side of the equation by using the FOIL method:
(2x - 3)^2 = (2x - 3)(2x - 3) = 4x^2 - 12x + 9
Now, our equation becomes:
4x^2 - 12x + 9 = 4x
Rearranging the Equation
To solve for x, we need to bring all the terms to one side:
4x^2 - 16x + 9 = 0
Solving the Quadratic Equation
We now have a quadratic equation. There are several ways to solve this:
- Factoring: In this case, factoring might be tricky.
- Quadratic Formula: The most reliable method is using the quadratic formula:
x = [-b ± √(b^2 - 4ac)] / 2a
Where a = 4, b = -16, and c = 9.
Substituting the values into the formula:
x = [16 ± √((-16)^2 - 4 * 4 * 9)] / (2 * 4)
x = [16 ± √(256 - 144)] / 8
x = [16 ± √112] / 8
x = [16 ± 4√7] / 8
x = 2 ± (√7) / 2
Solutions
Therefore, the solutions to the equation (2x-3)^2 = 4x are:
- x = 2 + (√7) / 2
- x = 2 - (√7) / 2
These are the two distinct values of x that satisfy the original equation.