Question

use the quadratic formula to solve. express your answer in simplest form.
12v² - 9v - 25 = -4v

Explanation:

Step1: Rewrite in standard form

First, rewrite the equation $12v^{2}-9v - 25=-4v$ as $12v^{2}-9v + 4v-25 = 0$, which simplifies to $12v^{2}-5v - 25=0$.

Step2: Identify coefficients

For the quadratic equation $ax^{2}+bx + c = 0$, here $a = 12$, $b=-5$, and $c=-25$.

Step3: Apply quadratic formula

The quadratic formula is $v=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Substitute the values:
\[

$$\begin{align*} v&=\frac{-(-5)\pm\sqrt{(-5)^{2}-4\times12\times(-25)}}{2\times12}\\ &=\frac{5\pm\sqrt{25 + 1200}}{24}\\ &=\frac{5\pm\sqrt{1225}}{24}\\ &=\frac{5\pm35}{24} \end{align*}$$

\]

Step4: Find two solutions

For the plus - sign: $v=\frac{5 + 35}{24}=\frac{40}{24}=\frac{5}{3}$.
For the minus - sign: $v=\frac{5-35}{24}=\frac{-30}{24}=-\frac{5}{4}$.

Answer:

$v=\frac{5}{3},v =-\frac{5}{4}$