Question

question
use the quadratic formula to solve. express your answer in simplest form.
19v² - v - 2 = 4v²
answer attempt 1 out of 2
additional solution no solution

Explanation:

Step1: Rearrange the equation

Subtract \(4v^{2}\) from both sides to get the quadratic equation in standard form \(ax^{2}+bx + c = 0\).
\(19v^{2}-v - 2-4v^{2}=0\)
Simplify to \(15v^{2}-v - 2 = 0\), where \(a = 15\), \(b=- 1\), \(c=-2\).

Step2: Apply the quadratic formula

The quadratic formula is \(v=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-1)^{2}-4\times15\times(-2)=1 + 120 = 121\).
Then, substitute \(a = 15\), \(b=-1\), \(\Delta = 121\) into the formula:
\(v=\frac{-(-1)\pm\sqrt{121}}{2\times15}=\frac{1\pm11}{30}\)

Step3: Find the two solutions

For the plus sign: \(v=\frac{1 + 11}{30}=\frac{12}{30}=\frac{2}{5}\)
For the minus sign: \(v=\frac{1-11}{30}=\frac{-10}{30}=-\frac{1}{3}\)

Answer:

\(v=\frac{2}{5}\) or \(v =-\frac{1}{3}\)