Enter the correct answer in the box.

Solve the equation [tex]$x^2 - 16x + 54 = 0$[/tex] by completing the square.

Fill in the values of [tex]$a$[/tex] and [tex][tex]$b$[/tex][/tex] to complete the solutions.



Answer :

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]