To solve the quadratic equation [tex]\( x^2 - 12x + 36 = 90 \)[/tex], follow the steps below:
1. Set the equation to zero:
Start by rearranging the given equation into the standard form of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex].
[tex]\[
x^2 - 12x + 36 - 90 = 0
\][/tex]
2. Simplify the equation:
[tex]\[
x^2 - 12x - 54 = 0
\][/tex]
3. Solve the quadratic equation:
To find the roots of the quadratic equation [tex]\( x^2 - 12x - 54 = 0 \)[/tex], use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -12 \)[/tex], and [tex]\( c = -54 \)[/tex].
4. Determine the discriminant:
[tex]\[
b^2 - 4ac = (-12)^2 - 4(1)(-54)
\][/tex]
[tex]\[
b^2 - 4ac = 144 + 216
\][/tex]
[tex]\[
b^2 - 4ac = 360
\][/tex]
5. Substitute the values back into the quadratic formula:
[tex]\[
x = \frac{-(-12) \pm \sqrt{360}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{12 \pm \sqrt{360}}{2}
\][/tex]
6. Simplify the square root and the fraction:
[tex]\[
\sqrt{360} = \sqrt{36 \cdot 10} = 6\sqrt{10}
\][/tex]
Substitute this back into the equation:
[tex]\[
x = \frac{12 \pm 6\sqrt{10}}{2}
\][/tex]
[tex]\[
x = \frac{12}{2} \pm \frac{6\sqrt{10}}{2}
\][/tex]
[tex]\[
x = 6 \pm 3\sqrt{10}
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\( x^2 - 12x - 54 = 0 \)[/tex] are:
[tex]\[
x = 6 - 3\sqrt{10} \quad \text{and} \quad x = 6 + 3\sqrt{10}
\][/tex]
So the correct answer is:
[tex]\[
x = 6 \pm 3\sqrt{10}
\][/tex]
