(2x-3)^2=4x-6

3 min read Jun 16, 2024

Solving the Equation (2x-3)^2 = 4x-6

This article will guide you through solving the equation (2x-3)^2 = 4x-6. We will break down the steps and explain the concepts involved.

Step 1: Expand the Square

First, we need to expand the square on the left side of the equation. Remember the formula: (a-b)^2 = a^2 - 2ab + b^2.

Applying this to our equation:

(2x-3)^2 = (2x)^2 - 2(2x)(3) + (3)^2 = 4x^2 - 12x + 9

Now our equation becomes: 4x^2 - 12x + 9 = 4x - 6

Step 2: Move All Terms to One Side

To solve the equation, we need to bring all the terms to one side. Subtract 4x and add 6 to both sides:

4x^2 - 12x + 9 - 4x + 6 = 0 4x^2 - 16x + 15 = 0

Step 3: Solve the Quadratic Equation

We now have a quadratic equation in the form ax^2 + bx + c = 0. We can solve this using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In our equation, a = 4, b = -16, and c = 15. Substituting these values into the quadratic formula:

x = (16 ± √((-16)^2 - 4 * 4 * 15)) / (2 * 4) x = (16 ± √(256 - 240)) / 8 x = (16 ± √16) / 8 x = (16 ± 4) / 8

This gives us two possible solutions:

x = (16 + 4) / 8 = 20/8 = 5/2
x = (16 - 4) / 8 = 12/8 = 3/2

Conclusion

Therefore, the solutions to the equation (2x-3)^2 = 4x-6 are x = 5/2 and x = 3/2. You can verify these solutions by plugging them back into the original equation.