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

Answer 1

To solve the equation \(-7x^2 - 10 = -12x\), we first rearrange the equation into standard form.

Start by moving all terms to one side:

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

Next, we can multiply the entire equation by \(-1\) to make it easier to work with:

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

Now, we can use the quadratic formula to find the solutions:

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

Here, \(a = 7\), \(b = -12\), and \(c = 10\).

Calculate the discriminant (\(b^2 - 4ac\)):

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

Since the discriminant is negative, the solutions will be complex conjugates.

Now substitute into the quadratic formula:

\[
x = \frac{12 \pm \sqrt{-136}}{2 \cdot 7}
\]

Simplifying \(\sqrt{-136}\):

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

Inserting this back into the formula gives:

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

Simplifying further:

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

Thus, the solutions are:

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

The final answer, expressed as a conjugate pair of solutions:

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