Pre-Algebra Examples

$2 (n + 7) \cdot (4 n) = - 2 n + 14$

Step 1

Simplify $2 (n + 7) \cdot (4 n)$ .

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Step 1.1

Rewrite.

$0 + 0 + 2 (n + 7) \cdot (4 n) = - 2 n + 14$

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Apply the distributive property.

$(2 n + 2 \cdot 7) \cdot (4 n) = - 2 n + 14$

Step 1.2.2

Multiply $2$ by $7$ .

$(2 n + 14) \cdot (4 n) = - 2 n + 14$

Step 1.2.3

Apply the distributive property.

$2 n (4 n) + 14 (4 n) = - 2 n + 14$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Rewrite using the commutative property of multiplication.

$2 \cdot 4 n \cdot n + 14 (4 n) = - 2 n + 14$

Step 1.2.4.2

Multiply $4$ by $14$ .

$2 \cdot 4 n \cdot n + 56 n = - 2 n + 14$

Step 1.3

Simplify each term.

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Step 1.3.1

Multiply $n$ by $n$ by adding the exponents.

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Step 1.3.1.1

Move $n$ .

$2 \cdot 4 (n \cdot n) + 56 n = - 2 n + 14$

Step 1.3.1.2

Multiply $n$ by $n$ .

$2 \cdot 4 n^{2} + 56 n = - 2 n + 14$

Step 1.3.2

Multiply $2$ by $4$ .

$8 n^{2} + 56 n = - 2 n + 14$

Step 2

Move all terms containing $n$ to the left side of the equation.

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Step 2.1

Add $2 n$ to both sides of the equation.

$8 n^{2} + 56 n + 2 n = 14$

Step 2.2

Add $56 n$ and $2 n$ .

$8 n^{2} + 58 n = 14$

Step 3

Subtract $14$ from both sides of the equation.

$8 n^{2} + 58 n - 14 = 0$

Step 4

Factor $2$ out of $8 n^{2} + 58 n - 14$ .

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Step 4.1

Factor $2$ out of $8 n^{2}$ .

$2 (4 n^{2}) + 58 n - 14 = 0$

Step 4.2

Factor $2$ out of $58 n$ .

$2 (4 n^{2}) + 2 (29 n) - 14 = 0$

Step 4.3

Factor $2$ out of $- 14$ .

$2 (4 n^{2}) + 2 (29 n) + 2 (- 7) = 0$

Step 4.4

Factor $2$ out of $2 (4 n^{2}) + 2 (29 n)$ .

$2 (4 n^{2} + 29 n) + 2 (- 7) = 0$

Step 4.5

Factor $2$ out of $2 (4 n^{2} + 29 n) + 2 (- 7)$ .

$2 (4 n^{2} + 29 n - 7) = 0$

Step 5

Divide each term in $2 (4 n^{2} + 29 n - 7) = 0$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 (4 n^{2} + 29 n - 7) = 0$ by $2$ .

$\frac{2 (4 n^{2} + 29 n - 7)}{2} = \frac{0}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 (4 n^{2} + 29 n - 7)}{2} = \frac{0}{2}$

Step 5.2.1.2

Divide $4 n^{2} + 29 n - 7$ by $1$ .

$4 n^{2} + 29 n - 7 = \frac{0}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $2$ .

$4 n^{2} + 29 n - 7 = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 4$ , $b = 29$ , and $c = - 7$ into the quadratic formula and solve for $n$ .

$\frac{- 29 \pm \sqrt{29^{2} - 4 \cdot (4 \cdot - 7)}}{2 \cdot 4}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $29$ to the power of $2$ .

$n = \frac{- 29 \pm \sqrt{841 - 4 \cdot 4 \cdot - 7}}{2 \cdot 4}$

Step 8.1.2

Multiply $- 4 \cdot 4 \cdot - 7$ .

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Step 8.1.2.1

Multiply $- 4$ by $4$ .

$n = \frac{- 29 \pm \sqrt{841 - 16 \cdot - 7}}{2 \cdot 4}$

Step 8.1.2.2

Multiply $- 16$ by $- 7$ .

$n = \frac{- 29 \pm \sqrt{841 + 112}}{2 \cdot 4}$

Step 8.1.3

Add $841$ and $112$ .

$n = \frac{- 29 \pm \sqrt{953}}{2 \cdot 4}$

Step 8.2

Multiply $2$ by $4$ .

$n = \frac{- 29 \pm \sqrt{953}}{8}$

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 $29$ to the power of $2$ .

$n = \frac{- 29 \pm \sqrt{841 - 4 \cdot 4 \cdot - 7}}{2 \cdot 4}$

Step 9.1.2

Multiply $- 4 \cdot 4 \cdot - 7$ .

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Step 9.1.2.1

Multiply $- 4$ by $4$ .

$n = \frac{- 29 \pm \sqrt{841 - 16 \cdot - 7}}{2 \cdot 4}$

Step 9.1.2.2

Multiply $- 16$ by $- 7$ .

$n = \frac{- 29 \pm \sqrt{841 + 112}}{2 \cdot 4}$

Step 9.1.3

Add $841$ and $112$ .

$n = \frac{- 29 \pm \sqrt{953}}{2 \cdot 4}$

Step 9.2

Multiply $2$ by $4$ .

$n = \frac{- 29 \pm \sqrt{953}}{8}$

Step 9.3

Change the $\pm$ to $+$ .

$n = \frac{- 29 + \sqrt{953}}{8}$

Step 9.4

Rewrite $- 29$ as $- 1 (29)$ .

$n = \frac{- 1 \cdot 29 + \sqrt{953}}{8}$

Step 9.5

Factor $- 1$ out of $\sqrt{953}$ .

$n = \frac{- 1 \cdot 29 - 1 (- \sqrt{953})}{8}$

Step 9.6

Factor $- 1$ out of $- 1 (29) - 1 (- \sqrt{953})$ .

$n = \frac{- 1 (29 - \sqrt{953})}{8}$

Step 9.7

Move the negative in front of the fraction.

$n = - \frac{29 - \sqrt{953}}{8}$

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 $29$ to the power of $2$ .

$n = \frac{- 29 \pm \sqrt{841 - 4 \cdot 4 \cdot - 7}}{2 \cdot 4}$

Step 10.1.2

Multiply $- 4 \cdot 4 \cdot - 7$ .

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Step 10.1.2.1

Multiply $- 4$ by $4$ .

$n = \frac{- 29 \pm \sqrt{841 - 16 \cdot - 7}}{2 \cdot 4}$

Step 10.1.2.2

Multiply $- 16$ by $- 7$ .

$n = \frac{- 29 \pm \sqrt{841 + 112}}{2 \cdot 4}$

Step 10.1.3

Add $841$ and $112$ .

$n = \frac{- 29 \pm \sqrt{953}}{2 \cdot 4}$

Step 10.2

Multiply $2$ by $4$ .

$n = \frac{- 29 \pm \sqrt{953}}{8}$

Step 10.3

Change the $\pm$ to $-$ .

$n = \frac{- 29 - \sqrt{953}}{8}$

Step 10.4

Rewrite $- 29$ as $- 1 (29)$ .

$n = \frac{- 1 \cdot 29 - \sqrt{953}}{8}$

Step 10.5

Factor $- 1$ out of $- \sqrt{953}$ .

$n = \frac{- 1 \cdot 29 - (\sqrt{953})}{8}$

Step 10.6

Factor $- 1$ out of $- 1 (29) - (\sqrt{953})$ .

$n = \frac{- 1 (29 + \sqrt{953})}{8}$

Step 10.7

Move the negative in front of the fraction.

$n = - \frac{29 + \sqrt{953}}{8}$

Step 11

The final answer is the combination of both solutions.

$n = - \frac{29 - \sqrt{953}}{8}, - \frac{29 + \sqrt{953}}{8}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{29 - \sqrt{953}}{8}, - \frac{29 + \sqrt{953}}{8}$

Decimal Form:

$n = 0.23383726 \dots, - 7.48383726 \dots$