Solve for [tex]$x[tex]$[/tex] in the equation [tex]$[/tex]x^2 - 12x + 36 - 90 = 0$[/tex].

A. [tex]$x = 6 \pm 3\sqrt{10}$[/tex]
B. [tex]$x = 6 \pm 2\sqrt{7}$[/tex]
C. [tex]$x = 12 \pm 3\sqrt{22}$[/tex]
D. [tex]$x = 12 \pm 3\sqrt{10}$[/tex]



Answer :

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].