Pre-Algebra Examples

$42 - 11 (4 n + 4) = 5 (7 n + 2) \cdot (61 n)$

Step 1

Since $n$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$5 (7 n + 2) \cdot (61 n) = 42 - 11 (4 n + 4)$

Step 2

Simplify $5 (7 n + 2) \cdot (61 n)$ .

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Step 2.1

Rewrite.

$0 + 0 + 5 (7 n + 2) \cdot (61 n) = 42 - 11 (4 n + 4)$

Step 2.2

Simplify by multiplying through.

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Step 2.2.1

Apply the distributive property.

$(5 (7 n) + 5 \cdot 2) \cdot (61 n) = 42 - 11 (4 n + 4)$

Step 2.2.2

Multiply.

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Step 2.2.2.1

Multiply $7$ by $5$ .

$(35 n + 5 \cdot 2) \cdot (61 n) = 42 - 11 (4 n + 4)$

Step 2.2.2.2

Multiply $5$ by $2$ .

$(35 n + 10) \cdot (61 n) = 42 - 11 (4 n + 4)$

Step 2.2.3

Apply the distributive property.

$35 n (61 n) + 10 (61 n) = 42 - 11 (4 n + 4)$

Step 2.2.4

Simplify the expression.

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Step 2.2.4.1

Rewrite using the commutative property of multiplication.

$35 \cdot 61 n \cdot n + 10 (61 n) = 42 - 11 (4 n + 4)$

Step 2.2.4.2

Multiply $61$ by $10$ .

$35 \cdot 61 n \cdot n + 610 n = 42 - 11 (4 n + 4)$

Step 2.3

Simplify each term.

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Step 2.3.1

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

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Step 2.3.1.1

Move $n$ .

$35 \cdot 61 (n \cdot n) + 610 n = 42 - 11 (4 n + 4)$

Step 2.3.1.2

Multiply $n$ by $n$ .

$35 \cdot 61 n^{2} + 610 n = 42 - 11 (4 n + 4)$

Step 2.3.2

Multiply $35$ by $61$ .

$2135 n^{2} + 610 n = 42 - 11 (4 n + 4)$

Step 3

Simplify $42 - 11 (4 n + 4)$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

$2135 n^{2} + 610 n = 42 - 11 (4 n) - 11 \cdot 4$

Step 3.1.2

Multiply $4$ by $- 11$ .

$2135 n^{2} + 610 n = 42 - 44 n - 11 \cdot 4$

Step 3.1.3

Multiply $- 11$ by $4$ .

$2135 n^{2} + 610 n = 42 - 44 n - 44$

Step 3.2

Subtract $44$ from $42$ .

$2135 n^{2} + 610 n = - 44 n - 2$

Step 4

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

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Step 4.1

Add $44 n$ to both sides of the equation.

$2135 n^{2} + 610 n + 44 n = - 2$

Step 4.2

Add $610 n$ and $44 n$ .

$2135 n^{2} + 654 n = - 2$

Step 5

Add $2$ to both sides of the equation.

$2135 n^{2} + 654 n + 2 = 0$

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 $a = 2135$ , $b = 654$ , and $c = 2$ into the quadratic formula and solve for $n$ .

$\frac{- 654 \pm \sqrt{654^{2} - 4 \cdot (2135 \cdot 2)}}{2 \cdot 2135}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $654$ to the power of $2$ .

$n = \frac{- 654 \pm \sqrt{427716 - 4 \cdot 2135 \cdot 2}}{2 \cdot 2135}$

Step 8.1.2

Multiply $- 4 \cdot 2135 \cdot 2$ .

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Step 8.1.2.1

Multiply $- 4$ by $2135$ .

$n = \frac{- 654 \pm \sqrt{427716 - 8540 \cdot 2}}{2 \cdot 2135}$

Step 8.1.2.2

Multiply $- 8540$ by $2$ .

$n = \frac{- 654 \pm \sqrt{427716 - 17080}}{2 \cdot 2135}$

Step 8.1.3

Subtract $17080$ from $427716$ .

$n = \frac{- 654 \pm \sqrt{410636}}{2 \cdot 2135}$

Step 8.1.4

Rewrite $410636$ as $2^{2} \cdot 102659$ .

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Step 8.1.4.1

Factor $4$ out of $410636$ .

$n = \frac{- 654 \pm \sqrt{4 (102659)}}{2 \cdot 2135}$

Step 8.1.4.2

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

$n = \frac{- 654 \pm \sqrt{2^{2} \cdot 102659}}{2 \cdot 2135}$

Step 8.1.5

Pull terms out from under the radical.

$n = \frac{- 654 \pm 2 \sqrt{102659}}{2 \cdot 2135}$

Step 8.2

Multiply $2$ by $2135$ .

$n = \frac{- 654 \pm 2 \sqrt{102659}}{4270}$

Step 8.3

Simplify $\frac{- 654 \pm 2 \sqrt{102659}}{4270}$ .

$n = \frac{- 327 \pm \sqrt{102659}}{2135}$

Step 9

The final answer is the combination of both solutions.

$n = - \frac{327 - \sqrt{102659}}{2135}, - \frac{327 + \sqrt{102659}}{2135}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{327 - \sqrt{102659}}{2135}, - \frac{327 + \sqrt{102659}}{2135}$

Decimal Form:

$n = - 0.00308925 \dots, - 0.30323392 \dots$