Question

Solve the equation −7x2−10=−12x. Write your conjugate pair solution(s) as two values separated by a ± sign.(1 point)

Answer 1

To solve the equation \(-7x^2 - 10 = -12x\), we first rearrange it into standard quadratic form. We can move all terms to one side of the equation:

\[
-7x^2 + 12x - 10 = 0
\]

Next, let's multiply through by -1 to make the leading coefficient positive:

\[
7x^2 - 12x + 10 = 0
\]

Now we can use the quadratic formula, which is given by:

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

For our equation, the coefficients are:
- \(a = 7\)
- \(b = -12\)
- \(c = 10\)

Now we can calculate the discriminant \(b^2 - 4ac\):

\[
b^2 - 4ac = (-12)^2 - 4(7)(10) = 144 - 280 = -136
\]

Since the discriminant is negative, there will be two complex (conjugate) solutions. Now we can plug the values back into the quadratic formula:

\[
x = \frac{-(-12) \pm \sqrt{-136}}{2(7)} = \frac{12 \pm \sqrt{-136}}{14}
\]

We simplify \(\sqrt{-136}\):

\[
\sqrt{-136} = \sqrt{136}i = \sqrt{4 \cdot 34}i = 2\sqrt{34}i
\]

Now, substituting back into the formula:

\[
x = \frac{12 \pm 2\sqrt{34}i}{14} = \frac{6 \pm \sqrt{34}i}{7}
\]

Therefore, the conjugate pair solutions to the equation are:

\[
x = \frac{6}{7} \pm \frac{\sqrt{34}}{7} i
\]

This can be expressed as:

\[
\frac{6}{7} \pm \frac{\sqrt{34}}{7} i
\]

Thus, the final answer is:

\[
\frac{6}{7} \pm \frac{\sqrt{34}}{7} i
\]