Basic Math Examples

$(r - 3) (r + 9) = 12$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Apply the distributive property.

$r (r + 9) - 3 (r + 9) = 12$

Step 1.1.2

Apply the distributive property.

$r \cdot r + r \cdot 9 - 3 (r + 9) = 12$

Step 1.1.3

Apply the distributive property.

$r \cdot r + r \cdot 9 - 3 r - 3 \cdot 9 = 12$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $r$ by $r$ .

$r^{2} + r \cdot 9 - 3 r - 3 \cdot 9 = 12$

Step 1.2.1.2

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

$r^{2} + 9 \cdot r - 3 r - 3 \cdot 9 = 12$

Step 1.2.1.3

Multiply $- 3$ by $9$ .

$r^{2} + 9 r - 3 r - 27 = 12$

Step 1.2.2

Subtract $3 r$ from $9 r$ .

$r^{2} + 6 r - 27 = 12$

Step 2

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

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Step 2.1

Subtract $12$ from both sides of the equation.

$r^{2} + 6 r - 27 - 12 = 0$

Step 2.2

Subtract $12$ from $- 27$ .

$r^{2} + 6 r - 39 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 1$ , $b = 6$ , and $c = - 39$ into the quadratic formula and solve for $r$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot - 39)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$r = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 39}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 39$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$r = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 39}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 39$ .

$r = \frac{- 6 \pm \sqrt{36 + 156}}{2 \cdot 1}$

Step 5.1.3

Add $36$ and $156$ .

$r = \frac{- 6 \pm \sqrt{192}}{2 \cdot 1}$

Step 5.1.4

Rewrite $192$ as $8^{2} \cdot 3$ .

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Step 5.1.4.1

Factor $64$ out of $192$ .

$r = \frac{- 6 \pm \sqrt{64 (3)}}{2 \cdot 1}$

Step 5.1.4.2

Rewrite $64$ as $8^{2}$ .

$r = \frac{- 6 \pm \sqrt{8^{2} \cdot 3}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$r = \frac{- 6 \pm 8 \sqrt{3}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$r = \frac{- 6 \pm 8 \sqrt{3}}{2}$

Step 5.3

Simplify $\frac{- 6 \pm 8 \sqrt{3}}{2}$ .

$r = - 3 \pm 4 \sqrt{3}$

Step 6

The final answer is the combination of both solutions.

$r = - 3 + 4 \sqrt{3}, - 3 - 4 \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$r = - 3 + 4 \sqrt{3}, - 3 - 4 \sqrt{3}$

Decimal Form:

$r = 3.92820323 \dots, - 9.92820323 \dots$