Pre-Algebra Examples

$7 (r + 3) + 7 (r - 3) = 7 (r + 3) (r - 3)$

Step 1

Simplify $7 (r + 3) + 7 (r - 3)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$7 r + 7 \cdot 3 + 7 (r - 3) = 7 (r + 3) (r - 3)$

Step 1.1.2

Multiply $7$ by $3$ .

$7 r + 21 + 7 (r - 3) = 7 (r + 3) (r - 3)$

Step 1.1.3

Apply the distributive property.

$7 r + 21 + 7 r + 7 \cdot - 3 = 7 (r + 3) (r - 3)$

Step 1.1.4

Multiply $7$ by $- 3$ .

$7 r + 21 + 7 r - 21 = 7 (r + 3) (r - 3)$

Step 1.2

Simplify by adding terms.

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Step 1.2.1

Combine the opposite terms in $7 r + 21 + 7 r - 21$ .

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Step 1.2.1.1

Subtract $21$ from $21$ .

$7 r + 7 r + 0 = 7 (r + 3) (r - 3)$

Step 1.2.1.2

Add $7 r + 7 r$ and $0$ .

$7 r + 7 r = 7 (r + 3) (r - 3)$

Step 1.2.2

Add $7 r$ and $7 r$ .

$14 r = 7 (r + 3) (r - 3)$

Step 2

Simplify $7 (r + 3) (r - 3)$ .

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Step 2.1

Simplify by multiplying through.

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Step 2.1.1

Apply the distributive property.

$14 r = (7 r + 7 \cdot 3) (r - 3)$

Step 2.1.2

Multiply $7$ by $3$ .

$14 r = (7 r + 21) (r - 3)$

Step 2.2

Expand $(7 r + 21) (r - 3)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$14 r = 7 r (r - 3) + 21 (r - 3)$

Step 2.2.2

Apply the distributive property.

$14 r = 7 r \cdot r + 7 r \cdot - 3 + 21 (r - 3)$

Step 2.2.3

Apply the distributive property.

$14 r = 7 r \cdot r + 7 r \cdot - 3 + 21 r + 21 \cdot - 3$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $r$ by $r$ by adding the exponents.

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Step 2.3.1.1.1

Move $r$ .

$14 r = 7 (r \cdot r) + 7 r \cdot - 3 + 21 r + 21 \cdot - 3$

Step 2.3.1.1.2

Multiply $r$ by $r$ .

$14 r = 7 r^{2} + 7 r \cdot - 3 + 21 r + 21 \cdot - 3$

Step 2.3.1.2

Multiply $- 3$ by $7$ .

$14 r = 7 r^{2} - 21 r + 21 r + 21 \cdot - 3$

Step 2.3.1.3

Multiply $21$ by $- 3$ .

$14 r = 7 r^{2} - 21 r + 21 r - 63$

Step 2.3.2

Add $- 21 r$ and $21 r$ .

$14 r = 7 r^{2} + 0 - 63$

Step 2.3.3

Add $7 r^{2}$ and $0$ .

$14 r = 7 r^{2} - 63$

Step 3

Subtract $7 r^{2}$ from both sides of the equation.

$14 r - 7 r^{2} = - 63$

Step 4

Add $63$ to both sides of the equation.

$14 r - 7 r^{2} + 63 = 0$

Step 5

Factor $- 7$ out of $14 r - 7 r^{2} + 63$ .

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Step 5.1

Reorder $14 r$ and $- 7 r^{2}$ .

$- 7 r^{2} + 14 r + 63 = 0$

Step 5.2

Factor $- 7$ out of $- 7 r^{2}$ .

$- 7 r^{2} + 14 r + 63 = 0$

Step 5.3

Factor $- 7$ out of $14 r$ .

$- 7 r^{2} - 7 (- 2 r) + 63 = 0$

Step 5.4

Factor $- 7$ out of $63$ .

$- 7 r^{2} - 7 (- 2 r) - 7 \cdot - 9 = 0$

Step 5.5

Factor $- 7$ out of $- 7 (r^{2}) - 7 (- 2 r)$ .

$- 7 (r^{2} - 2 r) - 7 \cdot - 9 = 0$

Step 5.6

Factor $- 7$ out of $- 7 (r^{2} - 2 r) - 7 (- 9)$ .

$- 7 (r^{2} - 2 r - 9) = 0$

Step 6

Divide each term in $- 7 (r^{2} - 2 r - 9) = 0$ by $- 7$ and simplify.

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Step 6.1

Divide each term in $- 7 (r^{2} - 2 r - 9) = 0$ by $- 7$ .

$\frac{- 7 (r^{2} - 2 r - 9)}{- 7} = \frac{0}{- 7}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 7$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 7 (r^{2} - 2 r - 9)}{- 7} = \frac{0}{- 7}$

Step 6.2.1.2

Divide $r^{2} - 2 r - 9$ by $1$ .

$r^{2} - 2 r - 9 = \frac{0}{- 7}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $0$ by $- 7$ .

$r^{2} - 2 r - 9 = 0$

Step 7

Use the quadratic formula to find the solutions.

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

Step 8

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 9$ into the quadratic formula and solve for $r$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot - 9)}}{2 \cdot 1}$

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$r = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot - 9$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$r = \frac{2 \pm \sqrt{4 - 4 \cdot - 9}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $- 9$ .

$r = \frac{2 \pm \sqrt{4 + 36}}{2 \cdot 1}$

Step 9.1.3

Add $4$ and $36$ .

$r = \frac{2 \pm \sqrt{40}}{2 \cdot 1}$

Step 9.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 9.1.4.1

Factor $4$ out of $40$ .

$r = \frac{2 \pm \sqrt{4 (10)}}{2 \cdot 1}$

Step 9.1.4.2

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

$r = \frac{2 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Step 9.1.5

Pull terms out from under the radical.

$r = \frac{2 \pm 2 \sqrt{10}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$r = \frac{2 \pm 2 \sqrt{10}}{2}$

Step 9.3

Simplify $\frac{2 \pm 2 \sqrt{10}}{2}$ .

$r = 1 \pm \sqrt{10}$

Step 10

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 10.1

Simplify the numerator.

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Step 10.1.1

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

$r = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 10.1.2

Multiply $- 4 \cdot 1 \cdot - 9$ .

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Step 10.1.2.1

Multiply $- 4$ by $1$ .

$r = \frac{2 \pm \sqrt{4 - 4 \cdot - 9}}{2 \cdot 1}$

Step 10.1.2.2

Multiply $- 4$ by $- 9$ .

$r = \frac{2 \pm \sqrt{4 + 36}}{2 \cdot 1}$

Step 10.1.3

Add $4$ and $36$ .

$r = \frac{2 \pm \sqrt{40}}{2 \cdot 1}$

Step 10.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 10.1.4.1

Factor $4$ out of $40$ .

$r = \frac{2 \pm \sqrt{4 (10)}}{2 \cdot 1}$

Step 10.1.4.2

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

$r = \frac{2 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Step 10.1.5

Pull terms out from under the radical.

$r = \frac{2 \pm 2 \sqrt{10}}{2 \cdot 1}$

Step 10.2

Multiply $2$ by $1$ .

$r = \frac{2 \pm 2 \sqrt{10}}{2}$

Step 10.3

Simplify $\frac{2 \pm 2 \sqrt{10}}{2}$ .

$r = 1 \pm \sqrt{10}$

Step 10.4

Change the $\pm$ to $+$ .

$r = 1 + \sqrt{10}$

Step 11

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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

$r = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 11.1.2

Multiply $- 4 \cdot 1 \cdot - 9$ .

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Step 11.1.2.1

Multiply $- 4$ by $1$ .

$r = \frac{2 \pm \sqrt{4 - 4 \cdot - 9}}{2 \cdot 1}$

Step 11.1.2.2

Multiply $- 4$ by $- 9$ .

$r = \frac{2 \pm \sqrt{4 + 36}}{2 \cdot 1}$

Step 11.1.3

Add $4$ and $36$ .

$r = \frac{2 \pm \sqrt{40}}{2 \cdot 1}$

Step 11.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 11.1.4.1

Factor $4$ out of $40$ .

$r = \frac{2 \pm \sqrt{4 (10)}}{2 \cdot 1}$

Step 11.1.4.2

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

$r = \frac{2 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Step 11.1.5

Pull terms out from under the radical.

$r = \frac{2 \pm 2 \sqrt{10}}{2 \cdot 1}$

Step 11.2

Multiply $2$ by $1$ .

$r = \frac{2 \pm 2 \sqrt{10}}{2}$

Step 11.3

Simplify $\frac{2 \pm 2 \sqrt{10}}{2}$ .

$r = 1 \pm \sqrt{10}$

Step 11.4

Change the $\pm$ to $-$ .

$r = 1 - \sqrt{10}$

Step 12

The final answer is the combination of both solutions.

$r = 1 + \sqrt{10}, 1 - \sqrt{10}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$r = 1 + \sqrt{10}, 1 - \sqrt{10}$

Decimal Form:

$r = 4.16227766 \dots, - 2.16227766 \dots$