Solve Using the Quadratic Formula 28n^2=-96-184n

$28 n^{2} = - 96 - 184 n$

Step 1

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

Add $96$ to both sides of the equation.

$28 n^{2} + 96 = - 184 n$

Add $184 n$ to both sides of the equation.

$28 n^{2} + 96 + 184 n = 0$

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 $a = 28$ , $b = 184$ , and $c = 96$ into the quadratic formula and solve for $n$ .

$\frac{- 184 \pm \sqrt{184^{2} - 4 \cdot (28 \cdot 96)}}{2 \cdot 28}$

Step 4

Simplify.

Simplify the numerator.

Raise $184$ to the power of $2$ .

$n = \frac{- 184 \pm \sqrt{33856 - 4 \cdot 28 \cdot 96}}{2 \cdot 28}$

Multiply $- 4 \cdot 28 \cdot 96$ .

Multiply $- 4$ by $28$ .

$n = \frac{- 184 \pm \sqrt{33856 - 112 \cdot 96}}{2 \cdot 28}$

Multiply $- 112$ by $96$ .

$n = \frac{- 184 \pm \sqrt{33856 - 10752}}{2 \cdot 28}$

Subtract $10752$ from $33856$ .

$n = \frac{- 184 \pm \sqrt{23104}}{2 \cdot 28}$

Rewrite $23104$ as $152^{2}$ .

$n = \frac{- 184 \pm \sqrt{152^{2}}}{2 \cdot 28}$

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

$n = \frac{- 184 \pm 152}{2 \cdot 28}$

Multiply $2$ by $28$ .

$n = \frac{- 184 \pm 152}{56}$

Simplify $\frac{- 184 \pm 152}{56}$ .

$n = \frac{- 23 \pm 19}{7}$

Step 5

The final answer is the combination of both solutions.

$n = - \frac{4}{7}, - 6$