Solve v+9=-6(x-2)^2

$$v + 9 = - 6 ( x - 2 ) ^ { 2 }$$

Solve for v

$$v=-6\left(x-2\right)^{2}-9$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$v+9=-6\left(x^{2}-4x+4\right)$$

Use the distributive property to multiply -6 by x^{2}-4x+4.

$$v+9=-6x^{2}+24x-24$$

Subtract 9 from both sides.

$$v=-6x^{2}+24x-24-9$$

Subtract 9 from -24 to get -33.

$$v=-6x^{2}+24x-33$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{-6v-54}}{6}+2$$

$$x=\frac{\sqrt{-6v-54}}{6}+2$$

Solve for x

$$x=-\frac{\sqrt{-6v-54}}{6}+2$$

$$x=\frac{\sqrt{-6v-54}}{6}+2\text{, }v\leq -9$$