To solve the equation \(-7x^2 - 10 = -12x\), we first rearrange the equation to standard quadratic form:
\[ -7x^2 + 12x - 10 = 0 \]
To make the calculations easier, we can multiply the entire equation by -1:
\[ 7x^2 - 12x + 10 = 0 \]
Next, we will use the quadratic formula to find the solutions. The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, \(a = 7\), \(b = -12\), and \(c = 10\). We first 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, we will have complex conjugate solutions. Now let's calculate the solutions using the quadratic formula:
\[ x = \frac{-(-12) \pm \sqrt{-136}}{2 \cdot 7} \] \[ x = \frac{12 \pm \sqrt{-136}}{14} \]
We can simplify \(\sqrt{-136}\) as follows:
\[ \sqrt{-136} = \sqrt{136} \cdot i = \sqrt{4 \cdot 34} \cdot i = 2\sqrt{34} \cdot i \]
Substituting this back into the equation, we get:
\[ x = \frac{12 \pm 2\sqrt{34}i}{14} \] \[ x = \frac{12}{14} \pm \frac{2\sqrt{34}i}{14} \] \[ x = \frac{6}{7} \pm \frac{\sqrt{34}}{7}i \]
Thus, the conjugate pair solutions are:
\[ x = \frac{6}{7} \pm \frac{\sqrt{34}}{7}i \]
So the final answer is:
\[ x = \frac{6}{7} \pm \frac{\sqrt{34}}{7}i \]