Question

Solve the equation below. Write your conjugate pair solutions as two values separated by a ± sign. -5x^2 - 8 = -6x

Answer 1

To solve the equation \(-5x^2 - 8 = -6x\), we first rearrange it into standard form. We can do this by moving all terms to one side of the equation:

\[
-5x^2 + 6x - 8 = 0
\]

Next, we can multiply through by \(-1\) to make the leading coefficient positive:

\[
5x^2 - 6x + 8 = 0
\]

Now we will use the quadratic formula to solve for \(x\), which is given by:

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

In our equation, \(a = 5\), \(b = -6\), and \(c = 8\). First, we calculate the discriminant \(b^2 - 4ac\):

\[
b^2 = (-6)^2 = 36
\]
\[
4ac = 4 \times 5 \times 8 = 160
\]
\[
b^2 - 4ac = 36 - 160 = -124
\]

Since the discriminant is negative, we will have complex solutions. Now we substitute \(b\), the square root of the discriminant, and \(a\) into the quadratic formula:

\[
x = \frac{-(-6) \pm \sqrt{-124}}{2 \times 5}
\]
\[
x = \frac{6 \pm \sqrt{-124}}{10}
\]

Next, we simplify \(\sqrt{-124}\):

\[
\sqrt{-124} = \sqrt{124}i = \sqrt{4 \times 31}i = 2\sqrt{31}i
\]

Now, substituting this back into the quadratic formula gives:

\[
x = \frac{6 \pm 2\sqrt{31}i}{10}
\]

We simplify this fraction:

\[
x = \frac{6}{10} \pm \frac{2\sqrt{31}i}{10}
\]
\[
x = \frac{3}{5} \pm \frac{\sqrt{31}}{5}i
\]

The solutions can be presented as:

\[
x = \frac{3}{5} \pm \frac{\sqrt{31}}{5}i
\]

Thus, the final answer is:

\[
\frac{3}{5} \pm \frac{\sqrt{31}}{5}i
\]