(x-9)^2=2x(x-9)-19

2 min read Jun 17, 2024
(x-9)^2=2x(x-9)-19

Solving the Quadratic Equation: (x-9)^2 = 2x(x-9) - 19

This article will guide you through the steps of solving the quadratic equation (x-9)^2 = 2x(x-9) - 19.

1. Expanding and Simplifying the Equation

First, we need to expand the equation and simplify it to get a standard quadratic form.

  • Expand the left side: (x-9)^2 = x^2 - 18x + 81
  • Expand the right side: 2x(x-9) - 19 = 2x^2 - 18x - 19

Now we have: x^2 - 18x + 81 = 2x^2 - 18x - 19

2. Rearranging the Equation

To get all terms on one side, subtract x^2, -18x, and 81 from both sides:

  • 0 = 2x^2 - 18x - 19 - x^2 + 18x + 81
  • 0 = x^2 + 62

3. Solving for x

Now we have a simple quadratic equation in the form ax^2 + c = 0. To solve for x, we can:

  • Subtract 62 from both sides: x^2 = -62
  • Take the square root of both sides: x = ±√(-62)
  • Simplify the radical: x = ±√(62)i, where 'i' is the imaginary unit (√-1)

4. Solution

Therefore, the solutions to the equation (x-9)^2 = 2x(x-9) - 19 are:

x = √(62)i and x = -√(62)i

These solutions are imaginary numbers, indicating that the original equation does not have any real number solutions.

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