Finite Math Examples

$a (n) = (\frac{7 \sqrt{n}}{1 + 3 \sqrt{n}})$

Step 1

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

$\frac{7 \sqrt{n}}{1 + 3 \sqrt{n}} = a n$

Step 2

Cross multiply.

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Step 2.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

$a n \cdot (1 + 3 \sqrt{n}) = 7 \sqrt{n}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $a n \cdot (1 + 3 \sqrt{n})$ .

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Step 2.2.1.1

Apply the distributive property.

$a n \cdot 1 + a n (3 \sqrt{n}) = 7 \sqrt{n}$

Step 2.2.1.2

Simplify the expression.

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Step 2.2.1.2.1

Multiply $a$ by $1$ .

$a n + a n (3 \sqrt{n}) = 7 \sqrt{n}$

Step 2.2.1.2.2

Rewrite using the commutative property of multiplication.

$a n + 3 a n \sqrt{n} = 7 \sqrt{n}$

Step 3

Solve for $\sqrt{n}$ .

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Step 3.1

Subtract $7 \sqrt{n}$ from both sides of the equation.

$a n + 3 a n \sqrt{n} - 7 \sqrt{n} = 0$

Step 3.2

Subtract $a n$ from both sides of the equation.

$3 a n \sqrt{n} - 7 \sqrt{n} = - a n$

Step 3.3

Factor $\sqrt{n}$ out of $3 a n \sqrt{n} - 7 \sqrt{n}$ .

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Step 3.3.1

Factor $\sqrt{n}$ out of $3 a n \sqrt{n}$ .

$\sqrt{n} (3 a n) - 7 \sqrt{n} = - a n$

Step 3.3.2

Factor $\sqrt{n}$ out of $- 7 \sqrt{n}$ .

$\sqrt{n} (3 a n) + \sqrt{n} \cdot - 7 = - a n$

Step 3.3.3

Factor $\sqrt{n}$ out of $\sqrt{n} (3 a n) + \sqrt{n} \cdot - 7$ .

$\sqrt{n} (3 a n - 7) = - a n$

Step 3.4

Divide each term in $\sqrt{n} (3 a n - 7) = - a n$ by $3 a n - 7$ and simplify.

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Step 3.4.1

Divide each term in $\sqrt{n} (3 a n - 7) = - a n$ by $3 a n - 7$ .

$\frac{\sqrt{n} (3 a n - 7)}{3 a n - 7} = \frac{- a n}{3 a n - 7}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $3 a n - 7$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{\sqrt{n} (3 a n - 7)}{3 a n - 7} = \frac{- a n}{3 a n - 7}$

Step 3.4.2.1.2

Divide $\sqrt{n}$ by $1$ .

$\sqrt{n} = \frac{- a n}{3 a n - 7}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Move the negative in front of the fraction.

$\sqrt{n} = - \frac{a n}{3 a n - 7}$

Step 4

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{n}}^{2} = {(- \frac{a n}{3 a n - 7})}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{n}$ as $n^{\frac{1}{2}}$ .

${(n^{\frac{1}{2}})}^{2} = {(- \frac{a n}{3 a n - 7})}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(n^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Multiply the exponents in ${(n^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$n^{\frac{1}{2} \cdot 2} = {(- \frac{a n}{3 a n - 7})}^{2}$

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

$n^{\frac{1}{2} \cdot 2} = {(- \frac{a n}{3 a n - 7})}^{2}$

Step 5.2.1.1.2.2

Rewrite the expression.

$n^{1} = {(- \frac{a n}{3 a n - 7})}^{2}$

Step 5.2.1.2

Simplify.

$n = {(- \frac{a n}{3 a n - 7})}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(- \frac{a n}{3 a n - 7})}^{2}$ .

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Step 5.3.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.3.1.1.1

Apply the product rule to $- \frac{a n}{3 a n - 7}$ .

$n = {(- 1)}^{2} {(\frac{a n}{3 a n - 7})}^{2}$

Step 5.3.1.1.2

Apply the product rule to $\frac{a n}{3 a n - 7}$ .

$n = {(- 1)}^{2} \frac{{(a n)}^{2}}{{(3 a n - 7)}^{2}}$

Step 5.3.1.1.3

Apply the product rule to $a n$ .

$n = {(- 1)}^{2} \frac{a^{2} n^{2}}{{(3 a n - 7)}^{2}}$

Step 5.3.1.2

Raise $- 1$ to the power of $2$ .

$n = 1 \frac{a^{2} n^{2}}{{(3 a n - 7)}^{2}}$

Step 5.3.1.3

Multiply $\frac{a^{2} n^{2}}{{(3 a n - 7)}^{2}}$ by $1$ .

$n = \frac{a^{2} n^{2}}{{(3 a n - 7)}^{2}}$

Step 6

Solve for $n$ .

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Step 6.1

Find the LCD of the terms in the equation.

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Step 6.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, {(3 a n - 7)}^{2}$

Step 6.1.2

The LCM of one and any expression is the expression.

${(3 a n - 7)}^{2}$

Step 6.2

Multiply each term in $n = \frac{a^{2} n^{2}}{{(3 a n - 7)}^{2}}$ by ${(3 a n - 7)}^{2}$ to eliminate the fractions.

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Step 6.2.1

Multiply each term in $n = \frac{a^{2} n^{2}}{{(3 a n - 7)}^{2}}$ by ${(3 a n - 7)}^{2}$ .

$n {(3 a n - 7)}^{2} = \frac{a^{2} n^{2}}{{(3 a n - 7)}^{2}} {(3 a n - 7)}^{2}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Cancel the common factor of ${(3 a n - 7)}^{2}$ .

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Step 6.2.2.1.1

Cancel the common factor.

$n {(3 a n - 7)}^{2} = \frac{a^{2} n^{2}}{{(3 a n - 7)}^{2}} {(3 a n - 7)}^{2}$

Step 6.2.2.1.2

Rewrite the expression.

$n {(3 a n - 7)}^{2} = a^{2} n^{2}$

Step 6.3

Solve the equation.

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Step 6.3.1

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

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Step 6.3.1.1

Subtract $a^{2} n^{2}$ from both sides of the equation.

$n {(3 a n - 7)}^{2} - a^{2} n^{2} = 0$

Step 6.3.1.2

Simplify each term.

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Step 6.3.1.2.1

Rewrite ${(3 a n - 7)}^{2}$ as $(3 a n - 7) (3 a n - 7)$ .

$n ((3 a n - 7) (3 a n - 7)) - a^{2} n^{2} = 0$

Step 6.3.1.2.2

Expand $(3 a n - 7) (3 a n - 7)$ using the FOIL Method.

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Step 6.3.1.2.2.1

Apply the distributive property.

$n (3 a n (3 a n - 7) - 7 (3 a n - 7)) - a^{2} n^{2} = 0$

Step 6.3.1.2.2.2

Apply the distributive property.

$n (3 a n (3 a n) + 3 a n \cdot - 7 - 7 (3 a n - 7)) - a^{2} n^{2} = 0$

Step 6.3.1.2.2.3

Apply the distributive property.

$n (3 a n (3 a n) + 3 a n \cdot - 7 - 7 (3 a n) - 7 \cdot - 7) - a^{2} n^{2} = 0$

Step 6.3.1.2.3

Simplify and combine like terms.

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Step 6.3.1.2.3.1

Simplify each term.

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Step 6.3.1.2.3.1.1

Multiply $a$ by $a$ by adding the exponents.

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Step 6.3.1.2.3.1.1.1

Move $a$ .

$n (3 (a \cdot a) n (3 n) + 3 a n \cdot - 7 - 7 (3 a n) - 7 \cdot - 7) - a^{2} n^{2} = 0$

Step 6.3.1.2.3.1.1.2

Multiply $a$ by $a$ .

$n (3 a^{2} n (3 n) + 3 a n \cdot - 7 - 7 (3 a n) - 7 \cdot - 7) - a^{2} n^{2} = 0$

Step 6.3.1.2.3.1.2

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

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Step 6.3.1.2.3.1.2.1

Move $n$ .

$n (3 a^{2} (n \cdot n) \cdot 3 + 3 a n \cdot - 7 - 7 (3 a n) - 7 \cdot - 7) - a^{2} n^{2} = 0$

Step 6.3.1.2.3.1.2.2

Multiply $n$ by $n$ .

$n (3 a^{2} n^{2} \cdot 3 + 3 a n \cdot - 7 - 7 (3 a n) - 7 \cdot - 7) - a^{2} n^{2} = 0$

Step 6.3.1.2.3.1.3

Multiply $3$ by $3$ .

$n (9 a^{2} n^{2} + 3 a n \cdot - 7 - 7 (3 a n) - 7 \cdot - 7) - a^{2} n^{2} = 0$

Step 6.3.1.2.3.1.4

Multiply $- 7$ by $3$ .

$n (9 a^{2} n^{2} - 21 a n - 7 (3 a n) - 7 \cdot - 7) - a^{2} n^{2} = 0$

Step 6.3.1.2.3.1.5

Multiply $3$ by $- 7$ .

$n (9 a^{2} n^{2} - 21 a n - 21 (a n) - 7 \cdot - 7) - a^{2} n^{2} = 0$

Step 6.3.1.2.3.1.6

Multiply $- 7$ by $- 7$ .

$n (9 a^{2} n^{2} - 21 a n - 21 a n + 49) - a^{2} n^{2} = 0$

Step 6.3.1.2.3.2

Subtract $21 a n$ from $- 21 a n$ .

$n (9 a^{2} n^{2} - 42 a n + 49) - a^{2} n^{2} = 0$

Step 6.3.1.2.4

Apply the distributive property.

$n (9 a^{2} n^{2}) + n (- 42 a n) + n \cdot 49 - a^{2} n^{2} = 0$

Step 6.3.1.2.5

Simplify.

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Step 6.3.1.2.5.1

Rewrite using the commutative property of multiplication.

$9 n (a^{2} n^{2}) + n (- 42 a n) + n \cdot 49 - a^{2} n^{2} = 0$

Step 6.3.1.2.5.2

Rewrite using the commutative property of multiplication.

$9 n (a^{2} n^{2}) - 42 n (a n) + n \cdot 49 - a^{2} n^{2} = 0$

Step 6.3.1.2.5.3

Move $49$ to the left of $n$ .

$9 n (a^{2} n^{2}) - 42 n (a n) + 49 \cdot n - a^{2} n^{2} = 0$

Step 6.3.1.2.6

Simplify each term.

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Step 6.3.1.2.6.1

Multiply $n$ by $n^{2}$ by adding the exponents.

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Step 6.3.1.2.6.1.1

Move $n^{2}$ .

$9 (n^{2} n) a^{2} - 42 n (a n) + 49 \cdot n - a^{2} n^{2} = 0$

Step 6.3.1.2.6.1.2

Multiply $n^{2}$ by $n$ .

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Step 6.3.1.2.6.1.2.1

Raise $n$ to the power of $1$ .

$9 (n^{2} n^{1}) a^{2} - 42 n (a n) + 49 \cdot n - a^{2} n^{2} = 0$

Step 6.3.1.2.6.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$9 n^{2 + 1} a^{2} - 42 n (a n) + 49 \cdot n - a^{2} n^{2} = 0$

Step 6.3.1.2.6.1.3

Add $2$ and $1$ .

$9 n^{3} a^{2} - 42 n (a n) + 49 \cdot n - a^{2} n^{2} = 0$

Step 6.3.1.2.6.2

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

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Step 6.3.1.2.6.2.1

Move $n$ .

$9 n^{3} a^{2} - 42 (n \cdot n) a + 49 \cdot n - a^{2} n^{2} = 0$

Step 6.3.1.2.6.2.2

Multiply $n$ by $n$ .

$9 n^{3} a^{2} - 42 n^{2} a + 49 \cdot n - a^{2} n^{2} = 0$

$9 n^{3} a^{2} - 42 n^{2} a + 49 n - a^{2} n^{2} = 0$

Step 6.3.2

Factor $n$ out of $9 n^{3} a^{2} - 42 n^{2} a + 49 n - a^{2} n^{2}$ .

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Step 6.3.2.1

Factor $n$ out of $9 n^{3} a^{2}$ .

$n (9 n^{2} a^{2}) - 42 n^{2} a + 49 n - a^{2} n^{2} = 0$

Step 6.3.2.2

Factor $n$ out of $- 42 n^{2} a$ .

$n (9 n^{2} a^{2}) + n (- 42 n a) + 49 n - a^{2} n^{2} = 0$

Step 6.3.2.3

Factor $n$ out of $49 n$ .

$n (9 n^{2} a^{2}) + n (- 42 n a) + n \cdot 49 - a^{2} n^{2} = 0$

Step 6.3.2.4

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

$n (9 n^{2} a^{2}) + n (- 42 n a) + n \cdot 49 + n (- a^{2} n) = 0$

Step 6.3.2.5

Factor $n$ out of $n (9 n^{2} a^{2}) + n (- 42 n a)$ .

$n (9 n^{2} a^{2} - 42 n a) + n \cdot 49 + n (- a^{2} n) = 0$

Step 6.3.2.6

Factor $n$ out of $n (9 n^{2} a^{2} - 42 n a) + n \cdot 49$ .

$n (9 n^{2} a^{2} - 42 n a + 49) + n (- a^{2} n) = 0$

Step 6.3.2.7

Factor $n$ out of $n (9 n^{2} a^{2} - 42 n a + 49) + n (- a^{2} n)$ .

$n (9 n^{2} a^{2} - 42 n a + 49 - a^{2} n) = 0$

Step 6.3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$n = 0$

$9 n^{2} a^{2} - 42 n a + 49 - a^{2} n = 0$

Step 6.3.4

Set $n$ equal to $0$ .

$n = 0$

Step 6.3.5

Set $9 n^{2} a^{2} - 42 n a + 49 - a^{2} n$ equal to $0$ and solve for $n$ .

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Step 6.3.5.1

Set $9 n^{2} a^{2} - 42 n a + 49 - a^{2} n$ equal to $0$ .

$9 n^{2} a^{2} - 42 n a + 49 - a^{2} n = 0$

Step 6.3.5.2

Solve $9 n^{2} a^{2} - 42 n a + 49 - a^{2} n = 0$ for $n$ .

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Step 6.3.5.2.1

Use the quadratic formula to find the solutions.

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

Step 6.3.5.2.2

Substitute the values $a = 9 a^{2}$ , $b = - 42 a - a^{2}$ , and $c = 49$ into the quadratic formula and solve for $n$ .

$\frac{- (- 42 a - a^{2}) \pm \sqrt{{(- 42 a - a^{2})}^{2} - 4 \cdot (9 a^{2} \cdot 49)}}{2 (9 a^{2})}$

Step 6.3.5.2.3

Simplify.

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Step 6.3.5.2.3.1

Simplify the numerator.

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Step 6.3.5.2.3.1.1

Apply the distributive property.

$n = \frac{- (- 42 a) + a^{2} \pm \sqrt{{(- 42 a - a^{2})}^{2} - 4 \cdot (9 a^{2}) \cdot 49}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.2

Multiply $- 42$ by $- 1$ .

$n = \frac{42 a + a^{2} \pm \sqrt{{(- 42 a - a^{2})}^{2} - 4 \cdot (9 a^{2}) \cdot 49}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.3

Multiply $- - a^{2}$ .

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Step 6.3.5.2.3.1.3.1

Multiply $- 1$ by $- 1$ .

$n = \frac{42 a + 1 a^{2} \pm \sqrt{{(- 42 a - a^{2})}^{2} - 4 \cdot (9 a^{2}) \cdot 49}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.3.2

Multiply $a^{2}$ by $1$ .

$n = \frac{42 a + a^{2} \pm \sqrt{{(- 42 a - a^{2})}^{2} - 4 \cdot (9 a^{2}) \cdot 49}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.4

Rewrite $4 \cdot 9 a^{2} \cdot 49$ as ${(2 \cdot 3 a \cdot 7)}^{2}$ .

$n = \frac{42 a + a^{2} \pm \sqrt{{(- 42 a - a^{2})}^{2} - {(2 \cdot (3 a) \cdot 7)}^{2}}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - 42 a - a^{2}$ and $b = 2 \cdot 3 a \cdot 7$ .

$n = \frac{42 a + a^{2} \pm \sqrt{(- 42 a - a^{2} + 2 \cdot (3 a) \cdot 7) (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6

Simplify.

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Step 6.3.5.2.3.1.6.1

Factor $a$ out of $- 42 a - a^{2} + 2 \cdot 3 a \cdot 7$ .

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Step 6.3.5.2.3.1.6.1.1

Factor $a$ out of $- 42 a$ .

$n = \frac{42 a + a^{2} \pm \sqrt{(a \cdot - 42 - a^{2} + 2 \cdot (3 a) \cdot 7) (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.1.2

Factor $a$ out of $- a^{2}$ .

$n = \frac{42 a + a^{2} \pm \sqrt{(a \cdot - 42 + a (- a) + 2 \cdot (3 a) \cdot 7) (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.1.3

Factor $a$ out of $2 \cdot 3 a \cdot 7$ .

$n = \frac{42 a + a^{2} \pm \sqrt{(a \cdot - 42 + a (- a) + a (2 \cdot 3 \cdot 7)) (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.1.4

Factor $a$ out of $a \cdot - 42 + a (- a)$ .

$n = \frac{42 a + a^{2} \pm \sqrt{(a \cdot (- 42 - a) + a (2 \cdot 3 \cdot 7)) (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.1.5

Factor $a$ out of $a \cdot (- 42 - a) + a (2 \cdot 3 \cdot 7)$ .

$n = \frac{42 a + a^{2} \pm \sqrt{a (- 42 - a + 2 \cdot 3 \cdot 7) (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.2

Multiply $2 \cdot 3 \cdot 7$ .

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Step 6.3.5.2.3.1.6.2.1

Multiply $2$ by $3$ .

$n = \frac{42 a + a^{2} \pm \sqrt{a (- 42 - a + 6 \cdot 7) (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.2.2

Multiply $6$ by $7$ .

$n = \frac{42 a + a^{2} \pm \sqrt{a (- 42 - a + 42) (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.3

Add $- 42$ and $42$ .

$n = \frac{42 a + a^{2} \pm \sqrt{a (- a + 0) (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.4

Add $- a$ and $0$ .

$n = \frac{42 a + a^{2} \pm \sqrt{a \cdot (- 1 a (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7)))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.5

Combine exponents.

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Step 6.3.5.2.3.1.6.5.1

Factor out negative.

$n = \frac{42 a + a^{2} \pm \sqrt{- a \cdot a (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.5.2

Raise $a$ to the power of $1$ .

$n = \frac{42 a + a^{2} \pm \sqrt{- a \cdot a (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.5.3

Raise $a$ to the power of $1$ .

$n = \frac{42 a + a^{2} \pm \sqrt{- a \cdot a (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$n = \frac{42 a + a^{2} \pm \sqrt{- a^{1 + 1} (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.5.5

Add $1$ and $1$ .

$n = \frac{42 a + a^{2} \pm \sqrt{- a^{2} (- 42 a - a^{2} - (2 \cdot (3 a) \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.6

Combine exponents.

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Step 6.3.5.2.3.1.6.6.1

Multiply $2$ by $3$ .

$n = \frac{42 a + a^{2} \pm \sqrt{- a^{2} (- 42 a - a^{2} - (6 a \cdot 7))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.6.2

Multiply $7$ by $6$ .

$n = \frac{42 a + a^{2} \pm \sqrt{- a^{2} (- 42 a - a^{2} - (42 a))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.6.6.3

Multiply $42$ by $- 1$ .

$n = \frac{42 a + a^{2} \pm \sqrt{- a^{2} (- 42 a - a^{2} - 42 a)}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.7

Subtract $42 a$ from $- 42 a$ .

$n = \frac{42 a + a^{2} \pm \sqrt{- a^{2} (- a^{2} - 84 a)}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.8

Factor $a$ out of $- a^{2} - 84 a$ .

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Step 6.3.5.2.3.1.8.1

Factor $a$ out of $- a^{2}$ .

$n = \frac{42 a + a^{2} \pm \sqrt{- a^{2} (a (- a) - 84 a)}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.8.2

Factor $a$ out of $- 84 a$ .

$n = \frac{42 a + a^{2} \pm \sqrt{- a^{2} (a (- a) + a \cdot - 84)}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.8.3

Factor $a$ out of $a (- a) + a \cdot - 84$ .

$n = \frac{42 a + a^{2} \pm \sqrt{- a^{2} (a (- a - 84))}}{2 \cdot (9 a^{2})}$

$n = \frac{42 a + a^{2} \pm \sqrt{- a^{2} a (- a - 84)}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.9

Combine exponents.

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Step 6.3.5.2.3.1.9.1

Raise $a$ to the power of $1$ .

$n = \frac{42 a + a^{2} \pm \sqrt{- a \cdot a^{2} (- a - 84)}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.9.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$n = \frac{42 a + a^{2} \pm \sqrt{- a^{1 + 2} (- a - 84)}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.9.3

Add $1$ and $2$ .

$n = \frac{42 a + a^{2} \pm \sqrt{- a^{3} (- a - 84)}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.10

Rewrite $- a^{3} (- a - 84)$ as $a^{2} \cdot (- a (- a - 84))$ .

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Step 6.3.5.2.3.1.10.1

Factor out $a^{2}$ .

$n = \frac{42 a + a^{2} \pm \sqrt{- a^{2} a (- a - 84)}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.10.2

Reorder $- 1$ and $a^{2}$ .

$n = \frac{42 a + a^{2} \pm \sqrt{a^{2} \cdot (- 1 a (- a - 84))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.10.3

Add parentheses.

$n = \frac{42 a + a^{2} \pm \sqrt{a^{2} \cdot (- 1 (a (- a - 84)))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.10.4

Add parentheses.

$n = \frac{42 a + a^{2} \pm \sqrt{a^{2} \cdot (- a (- a - 84))}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.1.11

Pull terms out from under the radical.

$n = \frac{42 a + a^{2} \pm a \sqrt{- a (- a - 84)}}{2 \cdot (9 a^{2})}$

Step 6.3.5.2.3.2

Multiply $2$ by $9$ .

$n = \frac{42 a + a^{2} \pm a \sqrt{- a (- a - 84)}}{18 a^{2}}$

Step 6.3.5.2.4

The final answer is the combination of both solutions.

$n = \frac{42 + a + \sqrt{- a (- a - 84)}}{18 a}$