Solve Using the Quadratic Formula 7v^2-42=-35v

$7 v^{2} - 42 = - 35 v$

Step 1

Add $35 v$ to both sides of the equation.

$7 v^{2} - 42 + 35 v = 0$

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 $a = 7$ , $b = 35$ , and $c = - 42$ into the quadratic formula and solve for $v$ .

$\frac{- 35 \pm \sqrt{35^{2} - 4 \cdot (7 \cdot - 42)}}{2 \cdot 7}$

Step 4

Simplify.

Simplify the numerator.

Raise $35$ to the power of $2$ .

$v = \frac{- 35 \pm \sqrt{1225 - 4 \cdot 7 \cdot - 42}}{2 \cdot 7}$

Multiply $- 4 \cdot 7 \cdot - 42$ .

Multiply $- 4$ by $7$ .

$v = \frac{- 35 \pm \sqrt{1225 - 28 \cdot - 42}}{2 \cdot 7}$

Multiply $- 28$ by $- 42$ .

$v = \frac{- 35 \pm \sqrt{1225 + 1176}}{2 \cdot 7}$

Add $1225$ and $1176$ .

$v = \frac{- 35 \pm \sqrt{2401}}{2 \cdot 7}$

Rewrite $2401$ as $49^{2}$ .

$v = \frac{- 35 \pm \sqrt{49^{2}}}{2 \cdot 7}$

Pull terms out from under the radical, assuming positive real numbers.

$v = \frac{- 35 \pm 49}{2 \cdot 7}$

Multiply $2$ by $7$ .

$v = \frac{- 35 \pm 49}{14}$

Simplify $\frac{- 35 \pm 49}{14}$ .

$v = \frac{- 5 \pm 7}{2}$

Step 5

The final answer is the combination of both solutions.

$v = 1, - 6$