Question

What are the roots of the equation x, squared, minus, 12, x, plus, 37, equals, 0x
2
−12x+37=0 in simplest a+bi form?

Answer 1

To find the roots of the quadratic equation \( x^2 - 12x + 37 = 0 \), we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = -12 \), and \( c = 37 \).

First, we calculate the discriminant \( b^2 - 4ac \):

\[
b^2 - 4ac = (-12)^2 - 4 \cdot 1 \cdot 37 = 144 - 148 = -4
\]

The discriminant is negative, which means the roots will be complex numbers.

Now, applying the quadratic formula:

\[
x = \frac{-(-12) \pm \sqrt{-4}}{2 \cdot 1} = \frac{12 \pm \sqrt{-4}}{2}
\]

The square root of \(-4\) can be simplified:

\[
\sqrt{-4} = 2i
\]

Now substituting this back into the equation:

\[
x = \frac{12 \pm 2i}{2} = \frac{12}{2} \pm \frac{2i}{2} = 6 \pm i
\]

Thus, the roots of the equation \( x^2 - 12x + 37 = 0 \) in simplest \( a + bi \) form are:

\[
\boxed{6 + i \text{ and } 6 - i}
\]