Trigonometry Examples

$\sqrt{x + 39} = x + 2$

Step 1

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

${\sqrt{x + 39}}^{2} = {(x + 2)}^{2}$

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 $\sqrt{x + 39}$ as ${(x + 39)}^{\frac{1}{2}}$ .

${({(x + 39)}^{\frac{1}{2}})}^{2} = {(x + 2)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x + 39)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x + 39)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(x + 39)}^{\frac{1}{2} \cdot 2} = {(x + 2)}^{2}$

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.

${(x + 39)}^{\frac{1}{2} \cdot 2} = {(x + 2)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x + 39)}^{1} = {(x + 2)}^{2}$

Step 2.2.1.2

Simplify.

$x + 39 = {(x + 2)}^{2}$

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 $(x + 2) (x + 2)$ .

$x + 39 = (x + 2) (x + 2)$

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.

$x + 39 = x (x + 2) + 2 (x + 2)$

Step 2.3.1.2.2

Apply the distributive property.

$x + 39 = x \cdot x + x \cdot 2 + 2 (x + 2)$

Step 2.3.1.2.3

Apply the distributive property.

$x + 39 = x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2$

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 $x$ .

$x + 39 = x^{2} + x \cdot 2 + 2 x + 2 \cdot 2$

Step 2.3.1.3.1.2

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

$x + 39 = x^{2} + 2 \cdot x + 2 x + 2 \cdot 2$

Step 2.3.1.3.1.3

Multiply $2$ by $2$ .

$x + 39 = x^{2} + 2 x + 2 x + 4$

Step 2.3.1.3.2

Add $2 x$ and $2 x$ .

$x + 39 = x^{2} + 4 x + 4$

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.

$x^{2} + 4 x + 4 = x + 39$

Step 3.2

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

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Step 3.2.1

Subtract $x$ from both sides of the equation.

$x^{2} + 4 x + 4 - x = 39$

Step 3.2.2

Subtract $x$ from $4 x$ .

$x^{2} + 3 x + 4 = 39$

Step 3.3

Move all terms to the left side of the equation and simplify.

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Step 3.3.1

Subtract $39$ from both sides of the equation.

$x^{2} + 3 x + 4 - 39 = 0$

Step 3.3.2

Subtract $39$ from $4$ .

$x^{2} + 3 x - 35 = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = 1$ , $b = 3$ , and $c = - 35$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot - 35)}}{2 \cdot 1}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 35}}{2 \cdot 1}$

Step 3.6.1.2

Multiply $- 4 \cdot 1 \cdot - 35$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 35}}{2 \cdot 1}$

Step 3.6.1.2.2

Multiply $- 4$ by $- 35$ .

$x = \frac{- 3 \pm \sqrt{9 + 140}}{2 \cdot 1}$

Step 3.6.1.3

Add $9$ and $140$ .

$x = \frac{- 3 \pm \sqrt{149}}{2 \cdot 1}$

Step 3.6.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{149}}{2}$

Step 3.7

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{149}}{2}, - \frac{3 + \sqrt{149}}{2}$

Step 4

Exclude the solutions that do not make $\sqrt{x + 39} = x + 2$ true.

$x = - \frac{3 - \sqrt{149}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 - \sqrt{149}}{2}$

Decimal Form:

$x = 4.60327780 \dots$