Of course! Let's solve the equation [tex]\(x^2 - 16x + 54 = 0\)[/tex] by completing the square. Here is a detailed, step-by-step solution:
1. Start with the given equation:
[tex]\[
x^2 - 16x + 54 = 0
\][/tex]
2. Move the constant term to the other side of the equation:
[tex]\[
x^2 - 16x = -54
\][/tex]
3. Complete the square:
- Take half of the coefficient of the [tex]\(x\)[/tex] term, which is [tex]\(-16\)[/tex]. Half of [tex]\(-16\)[/tex] is [tex]\(-8\)[/tex].
- Square [tex]\(-8\)[/tex], which is [tex]\(64\)[/tex].
To complete the square, add [tex]\(64\)[/tex] to both sides of the equation:
[tex]\[
x^2 - 16x + 64 = -54 + 64
\][/tex]
This simplifies to:
[tex]\[
(x - 8)^2 = 10
\][/tex]
4. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x - 8 = \pm \sqrt{10}
\][/tex]
5. Isolate [tex]\( x \)[/tex] by adding 8 to both sides:
[tex]\[
x = 8 \pm \sqrt{10}
\][/tex]
Therefore, the solutions to the equation are:
[tex]\[
x_1 = 8 + \sqrt{10}
\][/tex]
[tex]\[
x_2 = 8 - \sqrt{10}
\][/tex]
Given the approximate value of [tex]\(\sqrt{10} \approx 3.1622776601683795\)[/tex]:
- [tex]\(x_1 = 8 + 3.1622776601683795 \approx 11.16227766016838\)[/tex]
- [tex]\(x_2 = 8 - 3.1622776601683795 \approx 4.83772233983162\)[/tex]
So, in summary, the solutions are:
[tex]\[
\boxed{(a \approx 11.16227766016838, b \approx 4.83772233983162)}
\][/tex]
