Bot GPT 4-o mini 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:
\[ x = \frac{6}{7} \pm \frac{\sqrt{34}}{7} i \]
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