Algebra Examples

$\frac{x}{\sqrt{x + 11}} - 1 = \frac{1}{\sqrt{x + 11}}$

Step 1

Solve for $\sqrt{x + 11}$ .

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Step 1.1

Add $1$ to both sides of the equation.

$\frac{x}{\sqrt{x + 11}} = \frac{1}{\sqrt{x + 11}} + 1$

Step 1.2

Find the LCD of the terms in the equation.

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Step 1.2.1

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

$\sqrt{x + 11}, \sqrt{x + 11}, 1$

Step 1.2.2

Since $\sqrt{x + 11}, \sqrt{x + 11}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1$ then find LCM for the variable part ${\sqrt{x + 11}}^{1}, {\sqrt{x + 11}}^{1}$ .

Step 1.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.2.5

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 1.2.6

The factor for ${\sqrt{x + 11}}^{1}$ is $\sqrt{x + 11}$ itself.

${\sqrt{x + 11}}^{1} = \sqrt{x + 11}$

$\sqrt{x + 11}$ occurs $1$ time.

Step 1.2.7

The LCM of ${\sqrt{x + 11}}^{1}, {\sqrt{x + 11}}^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$\sqrt{x + 11}$

Step 1.3

Multiply each term in $\frac{x}{\sqrt{x + 11}} = \frac{1}{\sqrt{x + 11}} + 1$ by $\sqrt{x + 11}$ to eliminate the fractions.

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Step 1.3.1

Multiply each term in $\frac{x}{\sqrt{x + 11}} = \frac{1}{\sqrt{x + 11}} + 1$ by $\sqrt{x + 11}$ .

$\frac{x}{\sqrt{x + 11}} \sqrt{x + 11} = \frac{1}{\sqrt{x + 11}} \sqrt{x + 11} + 1 \sqrt{x + 11}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $\sqrt{x + 11}$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{x}{\sqrt{x + 11}} \sqrt{x + 11} = \frac{1}{\sqrt{x + 11}} \sqrt{x + 11} + 1 \sqrt{x + 11}$

Step 1.3.2.1.2

Rewrite the expression.

$x = \frac{1}{\sqrt{x + 11}} \sqrt{x + 11} + 1 \sqrt{x + 11}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Cancel the common factor of $\sqrt{x + 11}$ .

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Step 1.3.3.1.1.1

Cancel the common factor.

$x = \frac{1}{\sqrt{x + 11}} \sqrt{x + 11} + 1 \sqrt{x + 11}$

Step 1.3.3.1.1.2

Rewrite the expression.

$x = 1 + 1 \sqrt{x + 11}$

Step 1.3.3.1.2

Multiply $\sqrt{x + 11}$ by $1$ .

$x = 1 + \sqrt{x + 11}$

Step 1.4

Solve the equation.

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Step 1.4.1

Rewrite the equation as $1 + \sqrt{x + 11} = x$ .

$1 + \sqrt{x + 11} = x$

Step 1.4.2

Subtract $1$ from both sides of the equation.

$\sqrt{x + 11} = x - 1$

Step 2

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

${\sqrt{x + 11}}^{2} = {(x - 1)}^{2}$

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 $\sqrt{x + 11}$ as ${(x + 11)}^{\frac{1}{2}}$ .

${({(x + 11)}^{\frac{1}{2}})}^{2} = {(x - 1)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(x + 11)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(x + 11)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(x + 11)}^{\frac{1}{2} \cdot 2} = {(x - 1)}^{2}$

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.

${(x + 11)}^{\frac{1}{2} \cdot 2} = {(x - 1)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(x + 11)}^{1} = {(x - 1)}^{2}$

Step 3.2.1.2

Simplify.

$x + 11 = {(x - 1)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$x + 11 = (x - 1) (x - 1)$

Step 3.3.1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$x + 11 = x (x - 1) - 1 (x - 1)$

Step 3.3.1.2.2

Apply the distributive property.

$x + 11 = x \cdot x + x \cdot - 1 - 1 (x - 1)$

Step 3.3.1.2.3

Apply the distributive property.

$x + 11 = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1$

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 $x$ by $x$ .

$x + 11 = x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1$

Step 3.3.1.3.1.2

Move $- 1$ to the left of $x$ .

$x + 11 = x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1$

Step 3.3.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

$x + 11 = x^{2} - x - 1 x - 1 \cdot - 1$

Step 3.3.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

$x + 11 = x^{2} - x - x - 1 \cdot - 1$

Step 3.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

$x + 11 = x^{2} - x - x + 1$

Step 3.3.1.3.2

Subtract $x$ from $- x$ .

$x + 11 = x^{2} - 2 x + 1$

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.

$x^{2} - 2 x + 1 = x + 11$

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.

$x^{2} - 2 x + 1 - x = 11$

Step 4.2.2

Subtract $x$ from $- 2 x$ .

$x^{2} - 3 x + 1 = 11$

Step 4.3

Subtract $11$ from both sides of the equation.

$x^{2} - 3 x + 1 - 11 = 0$

Step 4.4

Subtract $11$ from $1$ .

$x^{2} - 3 x - 10 = 0$

Step 4.5

Factor $x^{2} - 3 x - 10$ using the AC method.

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Step 4.5.1

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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 4.5.2

Write the factored form using these integers.

$(x - 5) (x + 2) = 0$

Step 4.6

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

$x - 5 = 0$

$x + 2 = 0$

Step 4.7

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 4.7.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4.7.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4.8

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

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Step 4.8.1

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

$x + 2 = 0$

Step 4.8.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4.9

The final solution is all the values that make $(x - 5) (x + 2) = 0$ true.

$x = 5, - 2$

Step 5

Exclude the solutions that do not make $\frac{x}{\sqrt{x + 11}} - 1 = \frac{1}{\sqrt{x + 11}}$ true.

$x = 5$