To solve the quadratic equation [tex]\( x^2 - 12x + 45 = 0 \)[/tex], follow these steps:
1. Identify the coefficients:
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. Here, we identify:
[tex]\[
a = 1, \quad b = -12, \quad c = 45
\][/tex]
2. Calculate the discriminant:
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the coefficients [tex]\( a = 1 \)[/tex], [tex]\( b = -12 \)[/tex], and [tex]\( c = 45 \)[/tex]:
[tex]\[
\Delta = (-12)^2 - 4 \times 1 \times 45 = 144 - 180 = -36
\][/tex]
Since the discriminant is negative, the quadratic equation has two complex roots.
3. Apply the quadratic formula:
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Since the discriminant [tex]\( \Delta = -36 \)[/tex], we have:
[tex]\[
\sqrt{\Delta} = \sqrt{-36} = 6i
\][/tex]
(Here, [tex]\( i \)[/tex] is the imaginary unit, defined as [tex]\( i = \sqrt{-1} \)[/tex]).
Substitute the values of [tex]\( b = -12 \)[/tex], [tex]\( \sqrt{\Delta} = 6i \)[/tex], and [tex]\( a = 1 \)[/tex] into the quadratic formula:
[tex]\[
x = \frac{12 \pm 6i}{2}
\][/tex]
4. Simplify each root:
Simplify the expression for each root:
[tex]\[
x = \frac{12 + 6i}{2} \quad \text{and} \quad x = \frac{12 - 6i}{2}
\][/tex]
This gives:
[tex]\[
x = 6 + 3i \quad \text{and} \quad x = 6 - 3i
\][/tex]
Therefore, the solutions to the equation [tex]\( x^2 - 12x + 45 = 0 \)[/tex] are [tex]\( x = 6 + 3i \)[/tex] and [tex]\( x = 6 - 3i \)[/tex].
The correct answer from the given options is:
[tex]\[
\boxed{B. \, x = 6 \pm 3i}
\][/tex]
