Solve Using the Quadratic Formula (r-3)(r+9)=-16

$(r - 3) (r + 9) = - 16$

Step 1

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

Simplify the left side.

Simplify $(r - 3) (r + 9)$ .

Expand $(r - 3) (r + 9)$ using the FOIL Method.

Apply the distributive property.

$r (r + 9) - 3 (r + 9) = - 16$

Apply the distributive property.

$r \cdot r + r \cdot 9 - 3 (r + 9) = - 16$

Apply the distributive property.

$r \cdot r + r \cdot 9 - 3 r - 3 \cdot 9 = - 16$

Simplify and combine like terms.

Simplify each term.

Multiply $r$ by $r$ .

$r^{2} + r \cdot 9 - 3 r - 3 \cdot 9 = - 16$

Move $9$ to the left of $r$ .

$r^{2} + 9 \cdot r - 3 r - 3 \cdot 9 = - 16$

Multiply $- 3$ by $9$ .

$r^{2} + 9 r - 3 r - 27 = - 16$

Subtract $3 r$ from $9 r$ .

$r^{2} + 6 r - 27 = - 16$

Add $16$ to both sides of the equation.

$r^{2} + 6 r - 27 + 16 = 0$

Add $- 27$ and $16$ .

$r^{2} + 6 r - 11 = 0$

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 $a = 1$ , $b = 6$ , and $c = - 11$ into the quadratic formula and solve for $r$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot - 11)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $6$ to the power of $2$ .

$r = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 11$ .

Multiply $- 4$ by $1$ .

$r = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4$ by $- 11$ .

$r = \frac{- 6 \pm \sqrt{36 + 44}}{2 \cdot 1}$

Add $36$ and $44$ .

$r = \frac{- 6 \pm \sqrt{80}}{2 \cdot 1}$

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

$r = \frac{- 6 \pm \sqrt{16 (5)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$r = \frac{- 6 \pm \sqrt{4^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$r = \frac{- 6 \pm 4 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$r = \frac{- 6 \pm 4 \sqrt{5}}{2}$

Simplify $\frac{- 6 \pm 4 \sqrt{5}}{2}$ .

$r = - 3 \pm 2 \sqrt{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$r = - 3 \pm 2 \sqrt{5}$

Decimal Form:

$r = 1.47213595 \dots, - 7.47213595 \dots$

Solve Using the Quadratic Formula (r-3)(r+9)=-16

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