Precalculus Examples

$\sqrt{x + 5} - 2 x = 5$

Step 1

Add $2 x$ to both sides of the equation.

$\sqrt{x + 5} = 5 + 2 x$

Step 2

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

${\sqrt{x + 5}}^{2} = {(5 + 2 x)}^{2}$

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 $\sqrt{x + 5}$ as ${(x + 5)}^{\frac{1}{2}}$ .

${({(x + 5)}^{\frac{1}{2}})}^{2} = {(5 + 2 x)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(x + 5)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(x + 5)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(x + 5)}^{\frac{1}{2} \cdot 2} = {(5 + 2 x)}^{2}$

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.

${(x + 5)}^{\frac{1}{2} \cdot 2} = {(5 + 2 x)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(x + 5)}^{1} = {(5 + 2 x)}^{2}$

Step 3.2.1.2

Simplify.

$x + 5 = {(5 + 2 x)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(5 + 2 x)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(5 + 2 x)}^{2}$ as $(5 + 2 x) (5 + 2 x)$ .

$x + 5 = (5 + 2 x) (5 + 2 x)$

Step 3.3.1.2

Expand $(5 + 2 x) (5 + 2 x)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$x + 5 = 5 (5 + 2 x) + 2 x (5 + 2 x)$

Step 3.3.1.2.2

Apply the distributive property.

$x + 5 = 5 \cdot 5 + 5 (2 x) + 2 x (5 + 2 x)$

Step 3.3.1.2.3

Apply the distributive property.

$x + 5 = 5 \cdot 5 + 5 (2 x) + 2 x \cdot 5 + 2 x (2 x)$

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 $5$ by $5$ .

$x + 5 = 25 + 5 (2 x) + 2 x \cdot 5 + 2 x (2 x)$

Step 3.3.1.3.1.2

Multiply $2$ by $5$ .

$x + 5 = 25 + 10 x + 2 x \cdot 5 + 2 x (2 x)$

Step 3.3.1.3.1.3

Multiply $5$ by $2$ .

$x + 5 = 25 + 10 x + 10 x + 2 x (2 x)$

Step 3.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$x + 5 = 25 + 10 x + 10 x + 2 \cdot 2 x \cdot x$

Step 3.3.1.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.3.1.5.1

Move $x$ .

$x + 5 = 25 + 10 x + 10 x + 2 \cdot 2 (x \cdot x)$

Step 3.3.1.3.1.5.2

Multiply $x$ by $x$ .

$x + 5 = 25 + 10 x + 10 x + 2 \cdot 2 x^{2}$

Step 3.3.1.3.1.6

Multiply $2$ by $2$ .

$x + 5 = 25 + 10 x + 10 x + 4 x^{2}$

Step 3.3.1.3.2

Add $10 x$ and $10 x$ .

$x + 5 = 25 + 20 x + 4 x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

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

$25 + 20 x + 4 x^{2} = x + 5$

Step 4.2

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

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Step 4.2.1

Subtract $x$ from both sides of the equation.

$25 + 20 x + 4 x^{2} - x = 5$

Step 4.2.2

Subtract $x$ from $20 x$ .

$25 + 4 x^{2} + 19 x = 5$

Step 4.3

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

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Step 4.3.1

Subtract $5$ from both sides of the equation.

$25 + 4 x^{2} + 19 x - 5 = 0$

Step 4.3.2

Subtract $5$ from $25$ .

$4 x^{2} + 19 x + 20 = 0$

Step 4.4

Use the quadratic formula to find the solutions.

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

Step 4.5

Substitute the values $a = 4$ , $b = 19$ , and $c = 20$ into the quadratic formula and solve for $x$ .

$\frac{- 19 \pm \sqrt{19^{2} - 4 \cdot (4 \cdot 20)}}{2 \cdot 4}$

Step 4.6

Simplify.

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

Raise $19$ to the power of $2$ .

$x = \frac{- 19 \pm \sqrt{361 - 4 \cdot 4 \cdot 20}}{2 \cdot 4}$

Step 4.6.1.2

Multiply $- 4 \cdot 4 \cdot 20$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 19 \pm \sqrt{361 - 16 \cdot 20}}{2 \cdot 4}$

Step 4.6.1.2.2

Multiply $- 16$ by $20$ .

$x = \frac{- 19 \pm \sqrt{361 - 320}}{2 \cdot 4}$

Step 4.6.1.3

Subtract $320$ from $361$ .

$x = \frac{- 19 \pm \sqrt{41}}{2 \cdot 4}$

Step 4.6.2

Multiply $2$ by $4$ .

$x = \frac{- 19 \pm \sqrt{41}}{8}$

Step 4.7

The final answer is the combination of both solutions.

$x = - \frac{19 - \sqrt{41}}{8}, - \frac{19 + \sqrt{41}}{8}$

Step 5

Exclude the solutions that do not make $\sqrt{x + 5} - 2 x = 5$ true.

$x = - \frac{19 - \sqrt{41}}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{19 - \sqrt{41}}{8}$

Decimal Form:

$x = - 1.57460947 \dots$