Solve for v v^2-7v-5=-6v

$v^{2} - 7 v - 5 = - 6 v$

Step 1

Move all terms containing $v$ to the left side of the equation.

Add $6 v$ to both sides of the equation.

$v^{2} - 7 v - 5 + 6 v = 0$

Add $- 7 v$ and $6 v$ .

$v^{2} - v - 5 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 1$ , $b = - 1$ , and $c = - 5$ into the quadratic formula and solve for $v$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$v = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 5$ .

Multiply $- 4$ by $1$ .

$v = \frac{1 \pm \sqrt{1 - 4 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4$ by $- 5$ .

$v = \frac{1 \pm \sqrt{1 + 20}}{2 \cdot 1}$

Add $1$ and $20$ .

$v = \frac{1 \pm \sqrt{21}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$v = \frac{1 \pm \sqrt{21}}{2}$

Step 5

The final answer is the combination of both solutions.

$v = \frac{1 + \sqrt{21}}{2}, \frac{1 - \sqrt{21}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$v = \frac{1 + \sqrt{21}}{2}, \frac{1 - \sqrt{21}}{2}$

Decimal Form:

$v = 2.79128784 \dots, - 1.79128784 \dots$