Precalculus Examples

$2 x - 5 = \sqrt{x + 4}$

Step 1

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

$\sqrt{x + 4} = 2 x - 5$

Step 2

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

${\sqrt{x + 4}}^{2} = {(2 x - 5)}^{2}$

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

${({(x + 4)}^{\frac{1}{2}})}^{2} = {(2 x - 5)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

${(x + 4)}^{\frac{1}{2} \cdot 2} = {(2 x - 5)}^{2}$

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.

${(x + 4)}^{\frac{1}{2} \cdot 2} = {(2 x - 5)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(x + 4)}^{1} = {(2 x - 5)}^{2}$

Step 3.2.1.2

Simplify.

$x + 4 = {(2 x - 5)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

$x + 4 = (2 x - 5) (2 x - 5)$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Apply the distributive property.

$x + 4 = 2 x (2 x - 5) - 5 (2 x - 5)$

Step 3.3.1.2.2

Apply the distributive property.

$x + 4 = 2 x (2 x) + 2 x \cdot - 5 - 5 (2 x - 5)$

Step 3.3.1.2.3

Apply the distributive property.

$x + 4 = 2 x (2 x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5$

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

Rewrite using the commutative property of multiplication.

$x + 4 = 2 \cdot 2 x \cdot x + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5$

Step 3.3.1.3.1.2

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

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Step 3.3.1.3.1.2.1

Move $x$ .

$x + 4 = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5$

Step 3.3.1.3.1.2.2

Multiply $x$ by $x$ .

$x + 4 = 2 \cdot 2 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5$

Step 3.3.1.3.1.3

Multiply $2$ by $2$ .

$x + 4 = 4 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5$

Step 3.3.1.3.1.4

Multiply $- 5$ by $2$ .

$x + 4 = 4 x^{2} - 10 x - 5 (2 x) - 5 \cdot - 5$

Step 3.3.1.3.1.5

Multiply $2$ by $- 5$ .

$x + 4 = 4 x^{2} - 10 x - 10 x - 5 \cdot - 5$

Step 3.3.1.3.1.6

Multiply $- 5$ by $- 5$ .

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

Step 3.3.1.3.2

Subtract $10 x$ from $- 10 x$ .

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

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.

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

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.

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

Step 4.2.2

Subtract $x$ from $- 20 x$ .

$4 x^{2} - 21 x + 25 = 4$

Step 4.3

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

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Step 4.3.1

Subtract $4$ from both sides of the equation.

$4 x^{2} - 21 x + 25 - 4 = 0$

Step 4.3.2

Subtract $4$ from $25$ .

$4 x^{2} - 21 x + 21 = 0$

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 $a = 4$ , $b = - 21$ , and $c = 21$ into the quadratic formula and solve for $x$ .

$\frac{21 \pm \sqrt{{(- 21)}^{2} - 4 \cdot (4 \cdot 21)}}{2 \cdot 4}$

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 $- 21$ to the power of $2$ .

$x = \frac{21 \pm \sqrt{441 - 4 \cdot 4 \cdot 21}}{2 \cdot 4}$

Step 4.6.1.2

Multiply $- 4 \cdot 4 \cdot 21$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{21 \pm \sqrt{441 - 16 \cdot 21}}{2 \cdot 4}$

Step 4.6.1.2.2

Multiply $- 16$ by $21$ .

$x = \frac{21 \pm \sqrt{441 - 336}}{2 \cdot 4}$

Step 4.6.1.3

Subtract $336$ from $441$ .

$x = \frac{21 \pm \sqrt{105}}{2 \cdot 4}$

Step 4.6.2

Multiply $2$ by $4$ .

$x = \frac{21 \pm \sqrt{105}}{8}$

Step 4.7

The final answer is the combination of both solutions.

$x = \frac{21 + \sqrt{105}}{8}, \frac{21 - \sqrt{105}}{8}$

Step 5

Exclude the solutions that do not make $2 x - 5 = \sqrt{x + 4}$ true.

$x = \frac{21 + \sqrt{105}}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{21 + \sqrt{105}}{8}$

Decimal Form:

$x = 3.90586884 \dots$