Question

Solve the quadratic equation by completing the square
x^2 - 12x + 22 = 0
First choose the appropriate form and fill in the blanks with the correct numbers
Then, solve the equation. Round your answer to the nearest hundredth. If there is more than one solution, separate them with commas.

Answer 1

To solve the quadratic equation \( x^2 - 12x + 22 = 0 \) by completing the square, we first rewrite the equation in the standard form \( (x - p)^2 = q \).

1. Start with the equation:
\[
x^2 - 12x + 22 = 0
\]

2. Move the constant term to the other side:
\[
x^2 - 12x = -22
\]

3. To complete the square, we need to take half of the coefficient of \( x \) (which is -12), square it, and add it to both sides. Half of -12 is -6, and squaring it gives us 36:
\[
x^2 - 12x + 36 = -22 + 36
\]

4. This simplifies to:
\[
x^2 - 12x + 36 = 14
\]

5. Now we can rewrite the left side as a perfect square:
\[
(x - 6)^2 = 14
\]

Now we solve for \( x \) by taking the square root of both sides:

6. Taking the square root, we get:
\[
x - 6 = \pm \sqrt{14}
\]

7. Solving for \( x \), we get two solutions:
\[
x = 6 + \sqrt{14} \quad \text{and} \quad x = 6 - \sqrt{14}
\]

Now, we can calculate the numeric values of these solutions:

8. Calculating:
\[
\sqrt{14} \approx 3.74166
\]

9. Thus, the solutions are:
\[
x_1 = 6 + \sqrt{14} \approx 6 + 3.74166 \approx 9.74166 \quad \text{(rounded to 2 decimal places: 9.74)}
\]
\[
x_2 = 6 - \sqrt{14} \approx 6 - 3.74166 \approx 2.25834 \quad \text{(rounded to 2 decimal places: 2.26)}
\]

Finally, we present the solutions:

**Answer**: \( x \approx 9.74, 2.26 \) (rounded to the nearest hundredth).