Question

solve for v.
\\( 9v^2 = -30v - 25 \\)
if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.
v =
(there are some input buttons on the right side of the input box, including fraction, mixed number, no solution, decimal, square root, number 5, multiplication sign buttons.)

Explanation:

Step1: Rearrange the equation

First, we move all terms to one side of the equation to get a quadratic equation in standard form \(ax^{2}+bx + c = 0\). So, we add \(30v\) and \(25\) to both sides of the equation \(9v^{2}=-30v - 25\).
We get \(9v^{2}+30v + 25 = 0\).

Step2: Identify coefficients

For the quadratic equation \(9v^{2}+30v + 25 = 0\), we can see that \(a = 9\), \(b = 30\), and \(c = 25\).

Step3: Use the quadratic formula or check for perfect square

We can check if this quadratic is a perfect square. The square of \(3v + 5\) is \((3v)^{2}+2\times3v\times5 + 5^{2}=9v^{2}+30v + 25\), which is exactly our quadratic equation. So, we can factor it as \((3v + 5)^{2}=0\).

Step4: Solve for v

To solve \((3v + 5)^{2}=0\), we take the square root of both sides. We get \(3v+5 = 0\) (since the square root of 0 is 0). Then, we solve for \(v\) by subtracting 5 from both sides and then dividing by 3.
Subtract 5: \(3v=-5\).
Divide by 3: \(v =-\frac{5}{3}\). Since the square has a multiplicity of 2, but both solutions are the same, so the solution is \(v =-\frac{5}{3}\).

Answer:

\(v =-\frac{5}{3}\)