Solve Using the Quadratic Formula 2n^2=-10n-7

$2 n^{2} = - 10 n - 7$

Step 1

Move all the expressions to the left side of the equation.

Add $10 n$ to both sides of the equation.

$2 n^{2} + 10 n = - 7$

Add $7$ to both sides of the equation.

$2 n^{2} + 10 n + 7 = 0$

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 $a = 2$ , $b = 10$ , and $c = 7$ into the quadratic formula and solve for $n$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (2 \cdot 7)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $10$ to the power of $2$ .

$n = \frac{- 10 \pm \sqrt{100 - 4 \cdot 2 \cdot 7}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 7$ .

Multiply $- 4$ by $2$ .

$n = \frac{- 10 \pm \sqrt{100 - 8 \cdot 7}}{2 \cdot 2}$

Multiply $- 8$ by $7$ .

$n = \frac{- 10 \pm \sqrt{100 - 56}}{2 \cdot 2}$

Subtract $56$ from $100$ .

$n = \frac{- 10 \pm \sqrt{44}}{2 \cdot 2}$

Rewrite $44$ as $2^{2} \cdot 11$ .

Factor $4$ out of $44$ .

$n = \frac{- 10 \pm \sqrt{4 (11)}}{2 \cdot 2}$

Rewrite $4$ as $2^{2}$ .

$n = \frac{- 10 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 2}$

Pull terms out from under the radical.

$n = \frac{- 10 \pm 2 \sqrt{11}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$n = \frac{- 10 \pm 2 \sqrt{11}}{4}$

Simplify $\frac{- 10 \pm 2 \sqrt{11}}{4}$ .

$n = \frac{- 5 \pm \sqrt{11}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$n = \frac{- 5 \pm \sqrt{11}}{2}$

Decimal Form:

$n = - 0.84168760 \dots, - 4.15831239 \dots$