Precalculus Examples

$\sqrt{3 n + 10} = n + 4$

Step 1

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

${\sqrt{3 n + 10}}^{2} = {(n + 4)}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 n + 10}$ as ${(3 n + 10)}^{\frac{1}{2}}$ .

${({(3 n + 10)}^{\frac{1}{2}})}^{2} = {(n + 4)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(3 n + 10)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(3 n + 10)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(3 n + 10)}^{\frac{1}{2} \cdot 2} = {(n + 4)}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(3 n + 10)}^{\frac{1}{2} \cdot 2} = {(n + 4)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(3 n + 10)}^{1} = {(n + 4)}^{2}$

Step 2.2.1.2

Simplify.

$3 n + 10 = {(n + 4)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

$3 n + 10 = (n + 4) (n + 4)$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Apply the distributive property.

$3 n + 10 = n (n + 4) + 4 (n + 4)$

Step 2.3.1.2.2

Apply the distributive property.

$3 n + 10 = n \cdot n + n \cdot 4 + 4 (n + 4)$

Step 2.3.1.2.3

Apply the distributive property.

$3 n + 10 = n \cdot n + n \cdot 4 + 4 n + 4 \cdot 4$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $n$ by $n$ .

$3 n + 10 = n^{2} + n \cdot 4 + 4 n + 4 \cdot 4$

Step 2.3.1.3.1.2

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

$3 n + 10 = n^{2} + 4 \cdot n + 4 n + 4 \cdot 4$

Step 2.3.1.3.1.3

Multiply $4$ by $4$ .

$3 n + 10 = n^{2} + 4 n + 4 n + 16$

Step 2.3.1.3.2

Add $4 n$ and $4 n$ .

$3 n + 10 = n^{2} + 8 n + 16$

Step 3

Solve for $n$ .

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Step 3.1

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

$n^{2} + 8 n + 16 = 3 n + 10$

Step 3.2

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

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Step 3.2.1

Subtract $3 n$ from both sides of the equation.

$n^{2} + 8 n + 16 - 3 n = 10$

Step 3.2.2

Subtract $3 n$ from $8 n$ .

$n^{2} + 5 n + 16 = 10$

Step 3.3

Subtract $10$ from both sides of the equation.

$n^{2} + 5 n + 16 - 10 = 0$

Step 3.4

Subtract $10$ from $16$ .

$n^{2} + 5 n + 6 = 0$

Step 3.5

Factor $n^{2} + 5 n + 6$ using the AC method.

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Step 3.5.1

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 $6$ and whose sum is $5$ .

$2, 3$

Step 3.5.2

Write the factored form using these integers.

$(n + 2) (n + 3) = 0$

Step 3.6

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

$n + 2 = 0$

$n + 3 = 0$

Step 3.7

Set $n + 2$ equal to $0$ and solve for $n$ .

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Step 3.7.1

Set $n + 2$ equal to $0$ .

$n + 2 = 0$

Step 3.7.2

Subtract $2$ from both sides of the equation.

$n = - 2$

Step 3.8

Set $n + 3$ equal to $0$ and solve for $n$ .

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Step 3.8.1

Set $n + 3$ equal to $0$ .

$n + 3 = 0$

Step 3.8.2

Subtract $3$ from both sides of the equation.

$n = - 3$

Step 3.9

The final solution is all the values that make $(n + 2) (n + 3) = 0$ true.

$n = - 2, - 3$