Question

Enter the correct answer in the box.
Solve the equation x2 − 16x + 54 = 0 by completing the square.

Fill in the values of a and b to complete the solutions.

Answer 1

To solve the equation \( x^2 - 16x + 54 = 0 \) by completing the square, we can follow these steps:

1. Start with the given equation:
\[
x^2 - 16x + 54 = 0
\]

2. Move the constant term to the right side:
\[
x^2 - 16x = -54
\]

3. To complete the square, we need to add and subtract \(\left(\frac{b}{2}\right)^2\) to the left side, where \(b\) is the coefficient of \(x\). Here, \(b = -16\), so:
\[
\left(\frac{-16}{2}\right)^2 = (-8)^2 = 64
\]

4. Add 64 to both sides:
\[
x^2 - 16x + 64 = -54 + 64
\]
\[
x^2 - 16x + 64 = 10
\]

5. Now, the left side is a perfect square trinomial:
\[
(x - 8)^2 = 10
\]

6. Take the square root of both sides:
\[
x - 8 = \pm \sqrt{10}
\]

7. Solve for \(x\):
\[
x = 8 \pm \sqrt{10}
\]

Thus, the solutions to the equation are:
\[
x = 8 + \sqrt{10} \quad \text{and} \quad x = 8 - \sqrt{10}
\]

If we needed to fill in specific values of \(a\) and \(b\) as per the final requirement, we can say:

- \(a = 8\)
- \(b = \sqrt{10}\)

Thus, you can enter:
- \(a = 8\)
- \(b = \sqrt{10}\)