To solve the equation \(x^2 - 12x + 36 - 90 = 0\), let's follow a systematic approach:
1. Simplify the given equation:
[tex]\[
x^2 - 12x + 36 - 90 = 0
\][/tex]
Combine the constant terms:
[tex]\[
x^2 - 12x - 54 = 0
\][/tex]
2. Identify the coefficients for the quadratic equation \(ax^2 + bx + c = 0\):
Here, \(a = 1\), \(b = -12\), and \(c = -54\).
3. Calculate the discriminant:
The discriminant \(D\) is given by the formula:
[tex]\[
D = b^2 - 4ac
\][/tex]
Plugging in the coefficients:
[tex]\[
D = (-12)^2 - 4(1)(-54)
\][/tex]
Simplify the expression:
[tex]\[
D = 144 + 216 = 360
\][/tex]
4. Find the roots using the quadratic formula:
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{D}}{2a}
\][/tex]
Substitute the values of \(a\), \(b\), and \(D\):
[tex]\[
x = \frac{12 \pm \sqrt{360}}{2}
\][/tex]
Simplify \(\sqrt{360}\):
[tex]\[
\sqrt{360} = \sqrt{36 \times 10} = 6\sqrt{10}
\][/tex]
Substitute back into the formula:
[tex]\[
x = \frac{12 \pm 6\sqrt{10}}{2}
\][/tex]
Simplify the expression:
[tex]\[
x = 6 \pm 3\sqrt{10}
\][/tex]
Thus, the solutions to the equation \(x^2 - 12x - 54 = 0\) are
[tex]\[
x = 6 + 3\sqrt{10} \quad \text{and} \quad x = 6 - 3\sqrt{10}
\][/tex]
Comparing these solutions to the given choices:
\( \boxed{1} \): \( x = 6 \pm 3\sqrt{10} \)
\(2\): \( x = 6 \pm 2\sqrt{7} \)
\(3\): \( x = 12 \pm 3\sqrt{22} \)
\(4\): \( x = 12 \pm 3\sqrt{10} \)
The correct solution corresponds to choice [tex]\(\boxed{1}\)[/tex]: [tex]\( x = 6 \pm 3\sqrt{10} \)[/tex].
