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 need to rearrange it into standard form \(ax^2 + bx + c = 0\).

Starting with the original equation:

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

This simplifies to:

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

Now we can rewrite the equation in standard form:

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

Here, \(a = 10\), \(b = 0\), and \(c = 8\).

Next, we will use the quadratic formula, which is given by:

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

Substituting the values of \(a\), \(b\), and \(c\) into the formula:

1. Calculate \(b^2 - 4ac\):

\[
b^2 - 4ac = 0^2 - 4(10)(8) = 0 - 320 = -320
\]

2. Now, substitute into the quadratic formula:

\[
x = \frac{-0 \pm \sqrt{-320}}{2(10)}
\]

This simplifies to:

\[
x = \frac{\pm \sqrt{-320}}{20}
\]

3. Since \(\sqrt{-320} = \sqrt{320}i = \sqrt{64 \cdot 5}i = 8\sqrt{5}i\), we can express the equation as:

\[
x = \frac{\pm 8\sqrt{5}i}{20}
\]

This simplifies to:

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

Thus, we can express the solutions as:

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

The final solutions in the form \(a \pm bi\) are:

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