Solve Using the Quadratic Formula n^2+13n+22=7

$n^{2} + 13 n + 22 = 7$

Step 1

Move all terms to the left side of the equation and simplify.

Subtract $7$ from both sides of the equation.

$n^{2} + 13 n + 22 - 7 = 0$

Subtract $7$ from $22$ .

$n^{2} + 13 n + 15 = 0$

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 $a = 1$ , $b = 13$ , and $c = 15$ into the quadratic formula and solve for $n$ .

$\frac{- 13 \pm \sqrt{13^{2} - 4 \cdot (1 \cdot 15)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $13$ to the power of $2$ .

$n = \frac{- 13 \pm \sqrt{169 - 4 \cdot 1 \cdot 15}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 15$ .

Multiply $- 4$ by $1$ .

$n = \frac{- 13 \pm \sqrt{169 - 4 \cdot 15}}{2 \cdot 1}$

Multiply $- 4$ by $15$ .

$n = \frac{- 13 \pm \sqrt{169 - 60}}{2 \cdot 1}$

Subtract $60$ from $169$ .

$n = \frac{- 13 \pm \sqrt{109}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$n = \frac{- 13 \pm \sqrt{109}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$n = \frac{- 13 \pm \sqrt{109}}{2}$

Decimal Form:

$n = - 1.27984674 \dots, - 11.72015325 \dots$