Precalculus Examples

$n + 4 = \sqrt{6 n + 40}$

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.

$\sqrt{6 n + 40} = n + 4$

Step 2

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

${\sqrt{6 n + 40}}^{2} = {(n + 4)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 n + 40}$ as ${(6 n + 40)}^{\frac{1}{2}}$ .

${({(6 n + 40)}^{\frac{1}{2}})}^{2} = {(n + 4)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(6 n + 40)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(6 n + 40)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(6 n + 40)}^{\frac{1}{2} \cdot 2} = {(n + 4)}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(6 n + 40)}^{\frac{1}{2} \cdot 2} = {(n + 4)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(6 n + 40)}^{1} = {(n + 4)}^{2}$

Step 3.2.1.2

Simplify.

$6 n + 40 = {(n + 4)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(n + 4)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(n + 4)}^{2}$ as $(n + 4) (n + 4)$ .

$6 n + 40 = (n + 4) (n + 4)$

Step 3.3.1.2

Expand $(n + 4) (n + 4)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$6 n + 40 = n (n + 4) + 4 (n + 4)$

Step 3.3.1.2.2

Apply the distributive property.

$6 n + 40 = n \cdot n + n \cdot 4 + 4 (n + 4)$

Step 3.3.1.2.3

Apply the distributive property.

$6 n + 40 = n \cdot n + n \cdot 4 + 4 n + 4 \cdot 4$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $n$ by $n$ .

$6 n + 40 = n^{2} + n \cdot 4 + 4 n + 4 \cdot 4$

Step 3.3.1.3.1.2

Move $4$ to the left of $n$ .

$6 n + 40 = n^{2} + 4 \cdot n + 4 n + 4 \cdot 4$

Step 3.3.1.3.1.3

Multiply $4$ by $4$ .

$6 n + 40 = n^{2} + 4 n + 4 n + 16$

Step 3.3.1.3.2

Add $4 n$ and $4 n$ .

$6 n + 40 = n^{2} + 8 n + 16$

Step 4

Solve for $n$ .

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Step 4.1

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

$n^{2} + 8 n + 16 = 6 n + 40$

Step 4.2

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

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Step 4.2.1

Subtract $6 n$ from both sides of the equation.

$n^{2} + 8 n + 16 - 6 n = 40$

Step 4.2.2

Subtract $6 n$ from $8 n$ .

$n^{2} + 2 n + 16 = 40$

Step 4.3

Subtract $40$ from both sides of the equation.

$n^{2} + 2 n + 16 - 40 = 0$

Step 4.4

Subtract $40$ from $16$ .

$n^{2} + 2 n - 24 = 0$

Step 4.5

Factor $n^{2} + 2 n - 24$ using the AC method.

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Step 4.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $2$ .

$- 4, 6$

Step 4.5.2

Write the factored form using these integers.

$(n - 4) (n + 6) = 0$

Step 4.6

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

$n - 4 = 0$

$n + 6 = 0$

Step 4.7

Set $n - 4$ equal to $0$ and solve for $n$ .

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Step 4.7.1

Set $n - 4$ equal to $0$ .

$n - 4 = 0$

Step 4.7.2

Add $4$ to both sides of the equation.

$n = 4$

Step 4.8

Set $n + 6$ equal to $0$ and solve for $n$ .

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Step 4.8.1

Set $n + 6$ equal to $0$ .

$n + 6 = 0$

Step 4.8.2

Subtract $6$ from both sides of the equation.

$n = - 6$

Step 4.9

The final solution is all the values that make $(n - 4) (n + 6) = 0$ true.

$n = 4, - 6$

Step 5

Exclude the solutions that do not make $n + 4 = \sqrt{6 n + 40}$ true.

$n = 4$