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(2x-7)^2-7(7-2x)=0

2 min read Jun 16, 2024
(2x-7)^2-7(7-2x)=0

Solving the Equation: (2x-7)^2 - 7(7-2x) = 0

This article will walk through the steps of solving the quadratic equation (2x-7)^2 - 7(7-2x) = 0.

Expanding the Equation

First, we need to expand the equation to simplify it. We can do this by using the following steps:

  1. Expand the square: (2x-7)^2 = (2x-7)(2x-7) = 4x^2 - 28x + 49
  2. Distribute the -7: -7(7-2x) = -49 + 14x

Now, the equation becomes: 4x^2 - 28x + 49 - 49 + 14x = 0

Simplifying the Equation

Combining like terms, we get:

4x^2 - 14x = 0

Solving for x

To solve for x, we can use the following methods:

  1. Factoring:

    • Factor out a 2x: 2x(2x - 7) = 0
    • Set each factor equal to zero: 2x = 0 or 2x - 7 = 0
    • Solve for x: x = 0 or x = 7/2

  2. Quadratic Formula:

    • Identify a, b, and c in the standard quadratic form (ax^2 + bx + c = 0): a = 4, b = -14, c = 0
    • Apply the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
    • Substitute the values and solve: x = (14 ± √(14^2 - 4 * 4 * 0)) / (2 * 4) = 14/8 = 7/4 or x = 0

Solutions

Therefore, the solutions to the equation (2x-7)^2 - 7(7-2x) = 0 are:

  • x = 0
  • x = 7/2

We can verify these solutions by plugging them back into the original equation and confirming that they make the equation true.

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