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