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

Question

x, squared, minus, 10, x, equals, minus, 22
x
2
−10x=
−22

Answer 1

To solve the equation \( x^2 - 10x = -22 \) by completing the square, we follow these steps:

1. Move the constant to the other side: \[ x^2 - 10x + 22 = 0 \]

2. Focus on the quadratic part \( x^2 - 10x \): To complete the square, we need to find the term that will complete it. Take half of the coefficient of \( x \), square it, and add it to both sides:

   • The coefficient of \( x \) is \(-10\). Half of that is \(-5\).
   • Squaring \(-5\) gives us \(25\).

3. Add and subtract \( 25 \) on the left side: \[ x^2 - 10x + 25 - 25 + 22 = 0 \] This simplifies to: \[ (x - 5)^2 - 3 = 0 \]

4. Isolate the square term: \[ (x - 5)^2 = 3 \]

5. Take the square root of both sides: \[ x - 5 = \pm \sqrt{3} \]

6. Solve for \( x \): \[ x = 5 \pm \sqrt{3} \]

Thus, the solutions are: \[ x = 5 + \sqrt{3} \quad \text{and} \quad x = 5 - \sqrt{3} \]

In simplest form, the solutions are: \[ \boxed{5 + \sqrt{3}} \quad \text{and} \quad \boxed{5 - \sqrt{3}} \]