Question

What are the roots of the equation x, squared, minus, 6, x, plus, 45, equals, 0x
2
−6x+45=0 in simplest a+bi form?

Answer 1

To find the roots of the quadratic equation \( x^2 - 6x + 45 = 0 \), we'll use the quadratic formula:

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

Here, \( a = 1 \), \( b = -6 \), and \( c = 45 \).

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

\[
b^2 - 4ac = (-6)^2 - 4(1)(45) = 36 - 180 = -144
\]

Since the discriminant is negative, the roots will be complex. Now, we can plug the values into the quadratic formula:

\[
x = \frac{-(-6) \pm \sqrt{-144}}{2(1)} = \frac{6 \pm \sqrt{-144}}{2}
\]

Next, we simplify \( \sqrt{-144} \):

\[
\sqrt{-144} = \sqrt{144} \cdot i = 12i
\]

Now substitute back into the formula:

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

Dividing both terms by 2 gives us:

\[
x = 3 \pm 6i
\]

Thus, the roots of the equation \( x^2 - 6x + 45 = 0 \) in simplest \( a + bi \) form are:

\[
x = 3 + 6i \quad \text{and} \quad x = 3 - 6i
\]