Trigonometry Examples

$\sqrt{3 x + 16} = x + 2$

Step 1

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

${\sqrt{3 x + 16}}^{2} = {(x + 2)}^{2}$

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 $\sqrt{3 x + 16}$ as ${(3 x + 16)}^{\frac{1}{2}}$ .

${({(3 x + 16)}^{\frac{1}{2}})}^{2} = {(x + 2)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(3 x + 16)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(3 x + 16)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(3 x + 16)}^{\frac{1}{2} \cdot 2} = {(x + 2)}^{2}$

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.

${(3 x + 16)}^{\frac{1}{2} \cdot 2} = {(x + 2)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(3 x + 16)}^{1} = {(x + 2)}^{2}$

Step 2.2.1.2

Simplify.

$3 x + 16 = {(x + 2)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(x + 2)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$3 x + 16 = (x + 2) (x + 2)$

Step 2.3.1.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$3 x + 16 = x (x + 2) + 2 (x + 2)$

Step 2.3.1.2.2

Apply the distributive property.

$3 x + 16 = x \cdot x + x \cdot 2 + 2 (x + 2)$

Step 2.3.1.2.3

Apply the distributive property.

$3 x + 16 = x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2$

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 $x$ by $x$ .

$3 x + 16 = x^{2} + x \cdot 2 + 2 x + 2 \cdot 2$

Step 2.3.1.3.1.2

Move $2$ to the left of $x$ .

$3 x + 16 = x^{2} + 2 \cdot x + 2 x + 2 \cdot 2$

Step 2.3.1.3.1.3

Multiply $2$ by $2$ .

$3 x + 16 = x^{2} + 2 x + 2 x + 4$

Step 2.3.1.3.2

Add $2 x$ and $2 x$ .

$3 x + 16 = x^{2} + 4 x + 4$

Step 3

Solve for $x$ .

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Step 3.1

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

$x^{2} + 4 x + 4 = 3 x + 16$

Step 3.2

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

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Step 3.2.1

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

$x^{2} + 4 x + 4 - 3 x = 16$

Step 3.2.2

Subtract $3 x$ from $4 x$ .

$x^{2} + x + 4 = 16$

Step 3.3

Subtract $16$ from both sides of the equation.

$x^{2} + x + 4 - 16 = 0$

Step 3.4

Subtract $16$ from $4$ .

$x^{2} + x - 12 = 0$

Step 3.5

Factor $x^{2} + x - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 3.5.2

Write the factored form using these integers.

$(x - 3) (x + 4) = 0$

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 $0$ .

$x - 3 = 0$

$x + 4 = 0$

Step 3.7

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 3.7.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 3.7.2

Add $3$ to both sides of the equation.

$x = 3$

Step 3.8

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 3.8.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 3.8.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 3.9

The final solution is all the values that make $(x - 3) (x + 4) = 0$ true.

$x = 3, - 4$

Step 4

Exclude the solutions that do not make $\sqrt{3 x + 16} = x + 2$ true.

$x = 3$