Question

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

Answer 1

To solve the quadratic equation \(10x^2 + 5x = 5x - 8\), we first need to rearrange it into standard form. Starting from the given equation:

\[
10x^2 + 5x - 5x + 8 = 0
\]
Simplifying this gives:

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

Next, we'll move 8 to the other side:

\[
10x^2 = -8
\]

Now, divide both sides by 10:

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

Now, we can take the square root of both sides. However, since the right side is negative, we will introduce \(i\) (the imaginary unit):

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

To express the solution in a standard form \(a \pm bi\), we can rationalize the denominator:

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

Thus, the solutions of the equation are:

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

Putting it into the form \(a \pm bi\):

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

Therefore, the solutions in the form \(a \pm bi\) are:

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