Precalculus Examples

$f (x) = \frac{2 x - \sqrt{x - 4}}{\sqrt{x + 6} - x}$

Step 1

Set the radicand in $\sqrt{x - 4}$ greater than or equal to $0$ to find where the expression is defined.

$x - 4 \geq 0$

Step 2

Add $4$ to both sides of the inequality.

$x \geq 4$

Step 3

Set the radicand in $\sqrt{x + 6}$ greater than or equal to $0$ to find where the expression is defined.

$x + 6 \geq 0$

Step 4

Subtract $6$ from both sides of the inequality.

$x \geq - 6$

Step 5

Set the denominator in $\frac{2 x - \sqrt{x - 4}}{\sqrt{x + 6} - x}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x + 6} - x = 0$

Step 6

Solve for $x$ .

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Step 6.1

Add $x$ to both sides of the equation.

$\sqrt{x + 6} = x$

Step 6.2

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

${\sqrt{x + 6}}^{2} = x^{2}$

Step 6.3

Simplify each side of the equation.

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Step 6.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 6}$ as ${(x + 6)}^{\frac{1}{2}}$ .

${({(x + 6)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Simplify ${({(x + 6)}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.1.1

Multiply the exponents in ${({(x + 6)}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.1.1.1

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

${(x + 6)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 6.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.2.1.1.2.1

Cancel the common factor.

${(x + 6)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 6.3.2.1.1.2.2

Rewrite the expression.

${(x + 6)}^{1} = x^{2}$

Step 6.3.2.1.2

Simplify.

$x + 6 = x^{2}$

Step 6.4

Solve for $x$ .

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Step 6.4.1

Subtract $x^{2}$ from both sides of the equation.

$x + 6 - x^{2} = 0$

Step 6.4.2

Factor the left side of the equation.

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Step 6.4.2.1

Factor $- 1$ out of $x + 6 - x^{2}$ .

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Step 6.4.2.1.1

Reorder the expression.

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Step 6.4.2.1.1.1

Move $6$ .

$x - x^{2} + 6 = 0$

Step 6.4.2.1.1.2

Reorder $x$ and $- x^{2}$ .

$- x^{2} + x + 6 = 0$

Step 6.4.2.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + x + 6 = 0$

Step 6.4.2.1.3

Factor $- 1$ out of $x$ .

$- (x^{2}) - 1 (- x) + 6 = 0$

Step 6.4.2.1.4

Rewrite $6$ as $- 1 (- 6)$ .

$- (x^{2}) - 1 (- x) - 1 \cdot - 6 = 0$

Step 6.4.2.1.5

Factor $- 1$ out of $- (x^{2}) - 1 (- x)$ .

$- (x^{2} - x) - 1 \cdot - 6 = 0$

Step 6.4.2.1.6

Factor $- 1$ out of $- (x^{2} - x) - 1 (- 6)$ .

$- (x^{2} - x - 6) = 0$

Step 6.4.2.2

Factor.

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Step 6.4.2.2.1

Factor $x^{2} - x - 6$ using the AC method.

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Step 6.4.2.2.1.1

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 6.4.2.2.1.2

Write the factored form using these integers.

$- ((x - 3) (x + 2)) = 0$

Step 6.4.2.2.2

Remove unnecessary parentheses.

$- (x - 3) (x + 2) = 0$

Step 6.4.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 3 = 0$

$x + 2 = 0$

Step 6.4.4

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

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Step 6.4.4.1

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

$x - 3 = 0$

Step 6.4.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6.4.5

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 6.4.5.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 6.4.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6.4.6

The final solution is all the values that make $- (x - 3) (x + 2) = 0$ true.

$x = 3, - 2$

Step 6.5

Exclude the solutions that do not make $\sqrt{x + 6} - x = 0$ true.

$x = 3$

Step 7

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[4, \infty)$

Set-Builder Notation:

${x | x \geq 4}$

Step 8