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

Starting with the equation:

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

Now, we can simplify this by multiplying each term by \(-1\) to make the leading coefficient positive:

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

Next, we will use the quadratic formula to solve for \(x\):

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

Here, \(a = 7\), \(b = -12\), and \(c = 10\). Plugging in these values gives:

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

Since the discriminant (\(-136\)) is negative, we will have complex solutions. Now, we apply the quadratic formula:

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

This simplifies to:

\[
x = \frac{12 \pm \sqrt{-136}}{14}
\]

Now we simplify \(\sqrt{-136}\):

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

So we substitute this back into the equation:

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

We can simplify this further:

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

Thus, the conjugate pair solutions are:

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

Final answer:

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