Solve the equation for all values of x by completing the square. Express your answer in simplest form.

Question

x, squared, minus, 20, x, equals, minus, 90
x
2
−20x=
−90

Answer 1

To solve the equation \( x^2 - 20x = -90 \) by completing the square, we follow these steps:

1. Move the constant to the other side of the equation: \[ x^2 - 20x + 90 = 0 \]

2. Complete the square for the expression \( x^2 - 20x \): To complete the square, we take half of the coefficient of \( x \) (which is \(-20\)), square it, and add it to both sides. Half of \(-20\) is \(-10\), and squaring it gives \(100\).

3. Add and subtract \(100\) on the left side: \[ x^2 - 20x + 100 = 10 \] Now, we can rewrite the left side as a perfect square: \[ (x - 10)^2 = 10 \]

4. Take the square root of both sides: \[ x - 10 = \pm \sqrt{10} \]

5. Solve for \( x \): \[ x = 10 \pm \sqrt{10} \]

Thus, the solutions are: \[ x = 10 + \sqrt{10} \quad \text{and} \quad x = 10 - \sqrt{10} \]

In simplest form, the solutions to the equation \( x^2 - 20x = -90 \) are:

\[ \boxed{10 + \sqrt{10}} \quad \text{and} \quad \boxed{10 - \sqrt{10}} \]