Finite Math Examples

$\sqrt{x - 4} - \frac{2}{\sqrt{x - 4}} = 1$

Step 1

Solve for $\sqrt{x - 4}$ .

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Step 1.1

Find the LCD of the terms in the equation.

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Step 1.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, \sqrt{x - 4}, 1$

Step 1.1.2

The LCM of one and any expression is the expression.

$\sqrt{x - 4}$

Step 1.2

Multiply each term in $\sqrt{x - 4} - \frac{2}{\sqrt{x - 4}} = 1$ by $\sqrt{x - 4}$ to eliminate the fractions.

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Step 1.2.1

Multiply each term in $\sqrt{x - 4} - \frac{2}{\sqrt{x - 4}} = 1$ by $\sqrt{x - 4}$ .

$\sqrt{x - 4} \sqrt{x - 4} - \frac{2}{\sqrt{x - 4}} \sqrt{x - 4} = 1 \sqrt{x - 4}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

Multiply $\sqrt{x - 4}$ by $\sqrt{x - 4}$ .

${\sqrt{x - 4}}^{2} - \frac{2}{\sqrt{x - 4}} \sqrt{x - 4} = 1 \sqrt{x - 4}$

Step 1.2.2.1.2

Cancel the common factor of $\sqrt{x - 4}$ .

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Step 1.2.2.1.2.1

Move the leading negative in $- \frac{2}{\sqrt{x - 4}}$ into the numerator.

${\sqrt{x - 4}}^{2} + \frac{- 2}{\sqrt{x - 4}} \sqrt{x - 4} = 1 \sqrt{x - 4}$

Step 1.2.2.1.2.2

Cancel the common factor.

${\sqrt{x - 4}}^{2} + \frac{- 2}{\sqrt{x - 4}} \sqrt{x - 4} = 1 \sqrt{x - 4}$

Step 1.2.2.1.2.3

Rewrite the expression.

${\sqrt{x - 4}}^{2} - 2 = 1 \sqrt{x - 4}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Multiply $\sqrt{x - 4}$ by $1$ .

${\sqrt{x - 4}}^{2} - 2 = \sqrt{x - 4}$

Step 1.3

Solve the equation.

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Step 1.3.1

Subtract $\sqrt{x - 4}$ from both sides of the equation.

${\sqrt{x - 4}}^{2} - 2 - \sqrt{x - 4} = 0$

Step 1.3.2

Factor ${\sqrt{x - 4}}^{2} - 2 - \sqrt{x - 4}$ using the AC method.

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Step 1.3.2.1

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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 1.3.2.2

Write the factored form using these integers.

$(\sqrt{x - 4} - 2) (\sqrt{x - 4} + 1) = 0$

Step 1.3.3

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

$\sqrt{x - 4} - 2 = 0$

$\sqrt{x - 4} + 1 = 0$

Step 1.3.4

Set $\sqrt{x - 4} - 2$ equal to $0$ and solve for $\sqrt{x - 4}$ .

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Step 1.3.4.1

Set $\sqrt{x - 4} - 2$ equal to $0$ .

$\sqrt{x - 4} - 2 = 0$

Step 1.3.4.2

Add $2$ to both sides of the equation.

$\sqrt{x - 4} = 2$

Step 1.3.5

Set $\sqrt{x - 4} + 1$ equal to $0$ and solve for $\sqrt{x - 4}$ .

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Step 1.3.5.1

Set $\sqrt{x - 4} + 1$ equal to $0$ .

$\sqrt{x - 4} + 1 = 0$

Step 1.3.5.2

Subtract $1$ from both sides of the equation.

$\sqrt{x - 4} = - 1$

Step 1.3.6

The final solution is all the values that make $(\sqrt{x - 4} - 2) (\sqrt{x - 4} + 1) = 0$ true.

$\sqrt{x - 4} = 2, - 1$

Step 2

Solve for $x$ in $\sqrt{x - 4} = 2$ .

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Step 2.1

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

${\sqrt{x - 4}}^{2} = 2^{2}$

Step 2.2

Simplify each side of the equation.

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 4}$ as ${(x - 4)}^{\frac{1}{2}}$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

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

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Step 2.2.2.1.1

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

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Step 2.2.2.1.1.1

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

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

Step 2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.2.1.2

Simplify.

$x - 4 = 2^{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

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

$x - 4 = 4$

Step 2.3

Move all terms not containing $x$ to the right side of the equation.

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Step 2.3.1

Add $4$ to both sides of the equation.

$x = 4 + 4$

Step 2.3.2

Add $4$ and $4$ .

$x = 8$

Step 3

Solve for $x$ in $\sqrt{x - 4} = - 1$ .

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Step 3.1

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

${\sqrt{x - 4}}^{2} = {(- 1)}^{2}$

Step 3.2

Simplify each side of the equation.

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Step 3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 4}$ as ${(x - 4)}^{\frac{1}{2}}$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

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

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Step 3.2.2.1.1.1

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

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

Step 3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2.1.2

Simplify.

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Raise $- 1$ to the power of $2$ .

$x - 4 = 1$

Step 3.3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.3.1

Add $4$ to both sides of the equation.

$x = 1 + 4$

Step 3.3.2

Add $1$ and $4$ .

$x = 5$

Step 4

List all of the solutions.

$x = 8, 5$

Step 5

Exclude the solutions that do not make $\sqrt{x - 4} - \frac{2}{\sqrt{x - 4}} = 1$ true.

$x = 8$