Question

solve the equation for all real solutions in simplest form.\\(3v^2 + 10v + 1 = v\\)

Explanation:

Step1: Rearrange to standard quadratic form

Subtract $v$ from both sides:
$3v^2 + 10v + 1 - v = 0$
$3v^2 + 9v + 1 = 0$

Step2: Identify quadratic coefficients

For $ax^2+bx+c=0$, here $a=3$, $b=9$, $c=1$.

Step3: Apply quadratic formula

Quadratic formula: $v=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Substitute values:
$v=\frac{-9\pm\sqrt{9^2-4(3)(1)}}{2(3)}$

Step4: Simplify discriminant and result

Calculate discriminant: $\sqrt{81-12}=\sqrt{69}$
$v=\frac{-9\pm\sqrt{69}}{6}$

Answer:

$v=\frac{-9+\sqrt{69}}{6}$ and $v=\frac{-9-\sqrt{69}}{6}$