Pre-Algebra Examples

$n^{2} + {(n + 2)}^{2} = 156$

Step 1

Subtract $156$ from both sides of the equation.

$n^{2} + {(n + 2)}^{2} - 156 = 0$

Step 2

Simplify $n^{2} + {(n + 2)}^{2} - 156$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(n + 2)}^{2}$ as $(n + 2) (n + 2)$ .

$n^{2} + (n + 2) (n + 2) - 156 = 0$

Step 2.1.2

Expand $(n + 2) (n + 2)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$n^{2} + n (n + 2) + 2 (n + 2) - 156 = 0$

Step 2.1.2.2

Apply the distributive property.

$n^{2} + n \cdot n + n \cdot 2 + 2 (n + 2) - 156 = 0$

Step 2.1.2.3

Apply the distributive property.

$n^{2} + n \cdot n + n \cdot 2 + 2 n + 2 \cdot 2 - 156 = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $n$ by $n$ .

$n^{2} + n^{2} + n \cdot 2 + 2 n + 2 \cdot 2 - 156 = 0$

Step 2.1.3.1.2

Move $2$ to the left of $n$ .

$n^{2} + n^{2} + 2 \cdot n + 2 n + 2 \cdot 2 - 156 = 0$

Step 2.1.3.1.3

Multiply $2$ by $2$ .

$n^{2} + n^{2} + 2 n + 2 n + 4 - 156 = 0$

Step 2.1.3.2

Add $2 n$ and $2 n$ .

$n^{2} + n^{2} + 4 n + 4 - 156 = 0$

Step 2.2

Add $n^{2}$ and $n^{2}$ .

$2 n^{2} + 4 n + 4 - 156 = 0$

Step 2.3

Subtract $156$ from $4$ .

$2 n^{2} + 4 n - 152 = 0$

Step 3

Factor $2$ out of $2 n^{2} + 4 n - 152$ .

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Step 3.1

Factor $2$ out of $2 n^{2}$ .

$2 (n^{2}) + 4 n - 152 = 0$

Step 3.2

Factor $2$ out of $4 n$ .

$2 (n^{2}) + 2 (2 n) - 152 = 0$

Step 3.3

Factor $2$ out of $- 152$ .

$2 n^{2} + 2 (2 n) + 2 \cdot - 76 = 0$

Step 3.4

Factor $2$ out of $2 n^{2} + 2 (2 n)$ .

$2 (n^{2} + 2 n) + 2 \cdot - 76 = 0$

Step 3.5

Factor $2$ out of $2 (n^{2} + 2 n) + 2 \cdot - 76$ .

$2 (n^{2} + 2 n - 76) = 0$

Step 4

Divide each term in $2 (n^{2} + 2 n - 76) = 0$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 (n^{2} + 2 n - 76) = 0$ by $2$ .

$\frac{2 (n^{2} + 2 n - 76)}{2} = \frac{0}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 (n^{2} + 2 n - 76)}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide $n^{2} + 2 n - 76$ by $1$ .

$n^{2} + 2 n - 76 = \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $2$ .

$n^{2} + 2 n - 76 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = 2$ , and $c = - 76$ into the quadratic formula and solve for $n$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot - 76)}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$n = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 76}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot - 76$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$n = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 76}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $- 76$ .

$n = \frac{- 2 \pm \sqrt{4 + 304}}{2 \cdot 1}$

Step 7.1.3

Add $4$ and $304$ .

$n = \frac{- 2 \pm \sqrt{308}}{2 \cdot 1}$

Step 7.1.4

Rewrite $308$ as $2^{2} \cdot 77$ .

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Step 7.1.4.1

Factor $4$ out of $308$ .

$n = \frac{- 2 \pm \sqrt{4 (77)}}{2 \cdot 1}$

Step 7.1.4.2

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

$n = \frac{- 2 \pm \sqrt{2^{2} \cdot 77}}{2 \cdot 1}$

Step 7.1.5

Pull terms out from under the radical.

$n = \frac{- 2 \pm 2 \sqrt{77}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$n = \frac{- 2 \pm 2 \sqrt{77}}{2}$

Step 7.3

Simplify $\frac{- 2 \pm 2 \sqrt{77}}{2}$ .

$n = - 1 \pm \sqrt{77}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

$n = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 76}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot - 76$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$n = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 76}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $- 76$ .

$n = \frac{- 2 \pm \sqrt{4 + 304}}{2 \cdot 1}$

Step 8.1.3

Add $4$ and $304$ .

$n = \frac{- 2 \pm \sqrt{308}}{2 \cdot 1}$

Step 8.1.4

Rewrite $308$ as $2^{2} \cdot 77$ .

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Step 8.1.4.1

Factor $4$ out of $308$ .

$n = \frac{- 2 \pm \sqrt{4 (77)}}{2 \cdot 1}$

Step 8.1.4.2

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

$n = \frac{- 2 \pm \sqrt{2^{2} \cdot 77}}{2 \cdot 1}$

Step 8.1.5

Pull terms out from under the radical.

$n = \frac{- 2 \pm 2 \sqrt{77}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$n = \frac{- 2 \pm 2 \sqrt{77}}{2}$

Step 8.3

Simplify $\frac{- 2 \pm 2 \sqrt{77}}{2}$ .

$n = - 1 \pm \sqrt{77}$

Step 8.4

Change the $\pm$ to $+$ .

$n = - 1 + \sqrt{77}$

Step 9

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

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$n = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 76}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot - 76$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$n = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 76}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $- 76$ .

$n = \frac{- 2 \pm \sqrt{4 + 304}}{2 \cdot 1}$

Step 9.1.3

Add $4$ and $304$ .

$n = \frac{- 2 \pm \sqrt{308}}{2 \cdot 1}$

Step 9.1.4

Rewrite $308$ as $2^{2} \cdot 77$ .

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Step 9.1.4.1

Factor $4$ out of $308$ .

$n = \frac{- 2 \pm \sqrt{4 (77)}}{2 \cdot 1}$

Step 9.1.4.2

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

$n = \frac{- 2 \pm \sqrt{2^{2} \cdot 77}}{2 \cdot 1}$

Step 9.1.5

Pull terms out from under the radical.

$n = \frac{- 2 \pm 2 \sqrt{77}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$n = \frac{- 2 \pm 2 \sqrt{77}}{2}$

Step 9.3

Simplify $\frac{- 2 \pm 2 \sqrt{77}}{2}$ .

$n = - 1 \pm \sqrt{77}$

Step 9.4

Change the $\pm$ to $-$ .

$n = - 1 - \sqrt{77}$

Step 10

The final answer is the combination of both solutions.

$n = - 1 + \sqrt{77}, - 1 - \sqrt{77}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$n = - 1 + \sqrt{77}, - 1 - \sqrt{77}$

Decimal Form:

$n = 7.77496438 \dots, - 9.77496438 \dots$