Question

What are the roots of the equation x, squared, plus, 2, x, plus, 50, equals, 0x
2
+2x+50=0 in simplest a, plus, b, ia+bi form?

Answer 1

To find the roots of the equation \( x^2 + 2x + 50 = 0 \), we can use the quadratic formula:

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

Here, \( a = 1 \), \( b = 2 \), and \( c = 50 \).

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

\[
b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 50 = 4 - 200 = -196
\]

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

\[
x = \frac{{-2 \pm \sqrt{{-196}}}}{2 \cdot 1}
\]

The square root of \(-196\) can be calculated as follows:

\[
\sqrt{-196} = \sqrt{196} \cdot i = 14i
\]

Now substituting back into the formula:

\[
x = \frac{{-2 \pm 14i}}{2}
\]

This simplifies to:

\[
x = -1 \pm 7i
\]

Thus, the roots of the equation \( x^2 + 2x + 50 = 0 \) in the form \( a + bi \) are:

\[
x = -1 + 7i \quad \text{and} \quad x = -1 - 7i
\]