Question

solve for v.

$4v^2 = -7v - 3$

if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.

Explanation:

Step1: Rearrange the equation

First, we need to rewrite the given equation \(4v^{2}=-7v - 3\) in standard quadratic form \(ax^{2}+bx + c = 0\). We add \(7v\) and \(3\) to both sides of the equation:
\(4v^{2}+7v + 3=0\)

Step2: Use the quadratic formula

For a quadratic equation \(ax^{2}+bx + c = 0\), the quadratic formula is \(v=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). In our equation \(4v^{2}+7v + 3 = 0\), we have \(a = 4\), \(b=7\) and \(c = 3\).

First, we calculate the discriminant \(\Delta=b^{2}-4ac\). Substitute the values of \(a\), \(b\) and \(c\):
\(\Delta=(7)^{2}-4\times4\times3=49 - 48 = 1\)

Then we substitute into the quadratic formula:
\(v=\frac{-7\pm\sqrt{1}}{2\times4}=\frac{-7\pm1}{8}\)

Step3: Find the two solutions

We have two cases for the \(\pm\) sign:

Case 1: When we take the plus sign:
\(v=\frac{-7 + 1}{8}=\frac{-6}{8}=-\frac{3}{4}\)

Case 2: When we take the minus sign:
\(v=\frac{-7-1}{8}=\frac{-8}{8}=- 1\)

Answer:

\(-1,-\frac{3}{4}\)