Precalculus Examples

$1 + \sqrt{4 n + 28} = n$

Step 1

Subtract $1$ from both sides of the equation.

$\sqrt{4 n + 28} = n - 1$

Step 2

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

${\sqrt{4 n + 28}}^{2} = {(n - 1)}^{2}$

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 $\sqrt{4 n + 28}$ as ${(4 n + 28)}^{\frac{1}{2}}$ .

${({(4 n + 28)}^{\frac{1}{2}})}^{2} = {(n - 1)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(4 n + 28)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(4 n + 28)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(4 n + 28)}^{\frac{1}{2} \cdot 2} = {(n - 1)}^{2}$

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.

${(4 n + 28)}^{\frac{1}{2} \cdot 2} = {(n - 1)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(4 n + 28)}^{1} = {(n - 1)}^{2}$

Step 3.2.1.2

Simplify.

$4 n + 28 = {(n - 1)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(n - 1)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(n - 1)}^{2}$ as $(n - 1) (n - 1)$ .

$4 n + 28 = (n - 1) (n - 1)$

Step 3.3.1.2

Expand $(n - 1) (n - 1)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$4 n + 28 = n (n - 1) - 1 (n - 1)$

Step 3.3.1.2.2

Apply the distributive property.

$4 n + 28 = n \cdot n + n \cdot - 1 - 1 (n - 1)$

Step 3.3.1.2.3

Apply the distributive property.

$4 n + 28 = n \cdot n + n \cdot - 1 - 1 n - 1 \cdot - 1$

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 $n$ .

$4 n + 28 = n^{2} + n \cdot - 1 - 1 n - 1 \cdot - 1$

Step 3.3.1.3.1.2

Move $- 1$ to the left of $n$ .

$4 n + 28 = n^{2} - 1 \cdot n - 1 n - 1 \cdot - 1$

Step 3.3.1.3.1.3

Rewrite $- 1 n$ as $- n$ .

$4 n + 28 = n^{2} - n - 1 n - 1 \cdot - 1$

Step 3.3.1.3.1.4

Rewrite $- 1 n$ as $- n$ .

$4 n + 28 = n^{2} - n - n - 1 \cdot - 1$

Step 3.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

$4 n + 28 = n^{2} - n - n + 1$

Step 3.3.1.3.2

Subtract $n$ from $- n$ .

$4 n + 28 = n^{2} - 2 n + 1$

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.

$n^{2} - 2 n + 1 = 4 n + 28$

Step 4.2

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

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Step 4.2.1

Subtract $4 n$ from both sides of the equation.

$n^{2} - 2 n + 1 - 4 n = 28$

Step 4.2.2

Subtract $4 n$ from $- 2 n$ .

$n^{2} - 6 n + 1 = 28$

Step 4.3

Subtract $28$ from both sides of the equation.

$n^{2} - 6 n + 1 - 28 = 0$

Step 4.4

Subtract $28$ from $1$ .

$n^{2} - 6 n - 27 = 0$

Step 4.5

Factor $n^{2} - 6 n - 27$ 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 $- 27$ and whose sum is $- 6$ .

$- 9, 3$

Step 4.5.2

Write the factored form using these integers.

$(n - 9) (n + 3) = 0$

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 $0$ .

$n - 9 = 0$

$n + 3 = 0$

Step 4.7

Set $n - 9$ equal to $0$ and solve for $n$ .

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Step 4.7.1

Set $n - 9$ equal to $0$ .

$n - 9 = 0$

Step 4.7.2

Add $9$ to both sides of the equation.

$n = 9$

Step 4.8

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

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Step 4.8.1

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

$n + 3 = 0$

Step 4.8.2

Subtract $3$ from both sides of the equation.

$n = - 3$

Step 4.9

The final solution is all the values that make $(n - 9) (n + 3) = 0$ true.

$n = 9, - 3$

Step 5

Exclude the solutions that do not make $1 + \sqrt{4 n + 28} = n$ true.

$n = 9$