%5E2%3D2x(x-9)-19.jpeg "(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.