Question

solve for v.
$5v^2 = 19v + 4$
if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.
$v = \square$

Explanation:

Step1: Rearrange the equation

First, we rewrite the given equation \(5v^{2}=19v + 4\) in standard quadratic form \(ax^{2}+bx + c = 0\). Subtract \(19v\) and \(4\) from both sides:
\(5v^{2}-19v - 4=0\)

Step2: Factor the quadratic (or use quadratic formula)

We can factor the quadratic \(5v^{2}-19v - 4\). We need two numbers that multiply to \(5\times(- 4)=-20\) and add up to \(-19\). The numbers are \(-20\) and \(1\).
So we rewrite the middle term:
\(5v^{2}-20v + v-4 = 0\)
Factor by grouping:
\(5v(v - 4)+1(v - 4)=0\)
\((5v + 1)(v - 4)=0\)

Step3: Solve for \(v\)

Set each factor equal to zero:

• For \(5v+1 = 0\):

Subtract \(1\) from both sides: \(5v=-1\)
Divide by \(5\): \(v =-\frac{1}{5}\)

• For \(v - 4=0\):

Add \(4\) to both sides: \(v = 4\)

Answer:

\(v =-\frac{1}{5},4\)