Question

solve the equation $-7x^2 - 10 = -12x$. write your conjugate pair solution(s) as two values separated by a $\pm$ sign. (1 point)
$x = \square$

Explanation:

Step1: Rewrite in standard form

First, we rewrite the equation \(-7x^{2}-10 = -12x\) into standard quadratic form \(ax^{2}+bx + c=0\). We add \(12x\) to both sides and get \(-7x^{2}+12x - 10=0\). To make the coefficient of \(x^{2}\) positive, we multiply both sides by \(- 1\), so the equation becomes \(7x^{2}-12x + 10 = 0\). Here, \(a = 7\), \(b=-12\), \(c = 10\).

Step2: Use quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Substitute \(a = 7\), \(b=-12\), \(c = 10\) into the formula. First, calculate the discriminant \(\Delta=b^{2}-4ac=(-12)^{2}-4\times7\times10=144 - 280=- 136\). Then, \(x=\frac{-(-12)\pm\sqrt{-136}}{2\times7}=\frac{12\pm\sqrt{136}i}{14}\) (since \(\sqrt{-136}=\sqrt{136}\times\sqrt{-1}=\sqrt{4\times34}i = 2\sqrt{34}i\)). Simplify the fraction: \(\frac{12\pm2\sqrt{34}i}{14}=\frac{6\pm\sqrt{34}i}{7}\).

Answer:

\(\frac{6}{7}\pm\frac{\sqrt{34}}{7}i\)