Solve Using the Quadratic Formula n^2+5n=24

$n^{2} + 5 n = 24$

Step 1

Subtract $24$ from both sides of the equation.

$n^{2} + 5 n - 24 = 0$

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 $a = 1$ , $b = 5$ , and $c = - 24$ into the quadratic formula and solve for $n$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot - 24)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $5$ to the power of $2$ .

$n = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 24}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 24$ .

Multiply $- 4$ by $1$ .

$n = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 24}}{2 \cdot 1}$

Multiply $- 4$ by $- 24$ .

$n = \frac{- 5 \pm \sqrt{25 + 96}}{2 \cdot 1}$

Add $25$ and $96$ .

$n = \frac{- 5 \pm \sqrt{121}}{2 \cdot 1}$

Rewrite $121$ as $11^{2}$ .

$n = \frac{- 5 \pm \sqrt{11^{2}}}{2 \cdot 1}$

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

$n = \frac{- 5 \pm 11}{2 \cdot 1}$

Multiply $2$ by $1$ .

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

Step 5

The final answer is the combination of both solutions.

$n = 3, - 8$