Question

Use the quadratic formula to solve the quadratic equation 10x2+5x=5x−8 . Express its solutions in the form a±bi .(1 point)

Answer 1

To solve the quadratic equation \( 10x^2 + 5x = 5x - 8 \), we first rearrange the equation to bring all terms to one side:

\[
10x^2 + 5x - 5x + 8 = 0
\]

This simplifies the equation to:

\[
10x^2 + 8 = 0
\]

Next, we can isolate \( x^2 \) by moving \( 8 \) to the other side:

\[
10x^2 = -8
\]

Now, we divide both sides by \( 10 \):

\[
x^2 = -\frac{8}{10}
\]
\[
x^2 = -\frac{4}{5}
\]

Since we have a negative value under the square root, we will take the square root, leading us to complex solutions.

We can express \( x \) as follows:

\[
x = \pm \sqrt{-\frac{4}{5}}
\]

This can be rewritten using the imaginary unit \( i \):

\[
x = \pm \sqrt{\frac{4}{5}} \cdot i
\]
\[
x = \pm \frac{2}{\sqrt{5}} \cdot i
\]
\[
x = \pm \frac{2\sqrt{5}}{5} i
\]

Thus, the solutions of the equation \( 10x^2 + 8 = 0 \) expressed in the form \( a \pm bi \) are:

\[
0 \pm \frac{2\sqrt{5}}{5} i
\]

Finally, the solutions are:

\[
x = 0 + \frac{2\sqrt{5}}{5} i \quad \text{and} \quad x = 0 - \frac{2\sqrt{5}}{5} i
\]

In the final answer, we can express it as:

\[
\boxed{0 \pm \frac{2\sqrt{5}}{5} i}
\]