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

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

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.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $r$ by $r$ .

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

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

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

Multiply $- 3$ by $9$ .

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

Subtract $3 r$ from $9 r$ .

$r^{2} + 6 r - 27 = - 9$

Add $9$ to both sides of the equation.

$r^{2} + 6 r - 27 + 9 = 0$

Add $- 27$ and $9$ .

$r^{2} + 6 r - 18 = 0$

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 $c = - 18$ into the quadratic formula and solve for $r$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot - 18)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

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

$r = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 18}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 18$ .

Multiply $- 4$ by $1$ .

$r = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 18}}{2 \cdot 1}$

Multiply $- 4$ by $- 18$ .

$r = \frac{- 6 \pm \sqrt{36 + 72}}{2 \cdot 1}$

Add $36$ and $72$ .

$r = \frac{- 6 \pm \sqrt{108}}{2 \cdot 1}$

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$r = \frac{- 6 \pm \sqrt{36 (3)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$r = \frac{- 6 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$r = \frac{- 6 \pm 6 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$r = \frac{- 6 \pm 6 \sqrt{3}}{2}$

Simplify $\frac{- 6 \pm 6 \sqrt{3}}{2}$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$r = 2.19615242 \dots, - 8.19615242 \dots$

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

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