Question

solve the equation for all real solutions in simplest form.
$3v^2 - v - 12 = -6$

Explanation:

Step1: Rewrite in standard quadratic form

Add 6 to both sides to set equation to 0.
$3v^2 - v - 12 + 6 = 0$
$3v^2 - v - 6 = 0$

Step2: Identify coefficients for quadratic formula

For $ax^2+bx+c=0$, $a=3$, $b=-1$, $c=-6$.

Step3: Apply quadratic formula

Use $v=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
$v=\frac{-(-1)\pm\sqrt{(-1)^2-4(3)(-6)}}{2(3)}$

Step4: Simplify discriminant and numerator

Calculate discriminant and simplify terms.
$v=\frac{1\pm\sqrt{1+72}}{6}$
$v=\frac{1\pm\sqrt{73}}{6}$

Answer:

$v=\frac{1+\sqrt{73}}{6}$ and $v=\frac{1-\sqrt{73}}{6}$