Solve for v.v + 9 = 19v =

Answer 1

To solve the equation \( v \cdot v + 9 = 19v \), we start by rewriting it:

1. Rewrite the equation: \[ v^2 + 9 = 19v \]

2. Move all terms to one side: \[ v^2 - 19v + 9 = 0 \]

Now we can apply the quadratic formula, which is given by: \[ v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = -19 \), and \( c = 9 \).

3. Calculate the discriminant: \[ b^2 - 4ac = (-19)^2 - 4 \cdot 1 \cdot 9 = 361 - 36 = 325 \]

4. Plug the values into the quadratic formula: \[ v = \frac{-(-19) \pm \sqrt{325}}{2 \cdot 1} \] \[ v = \frac{19 \pm \sqrt{325}}{2} \]

5. Simplify \( \sqrt{325} \): \[ \sqrt{325} = \sqrt{25 \cdot 13} = 5\sqrt{13} \]

6. Substitute back into the equation: \[ v = \frac{19 \pm 5\sqrt{13}}{2} \]

Thus, the solutions for \( v \) are: \[ v = \frac{19 + 5\sqrt{13}}{2} \quad \text{and} \quad v = \frac{19 - 5\sqrt{13}}{2} \]