Algebra Examples

$\frac{x^{2} + 3}{x^{2} + 1} = 2$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

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

$x^{2} + 1, 1$

Step 1.2

Remove parentheses.

$x^{2} + 1, 1$

Step 1.3

The LCM of one and any expression is the expression.

$x^{2} + 1$

Step 2

Multiply each term in $\frac{x^{2} + 3}{x^{2} + 1} = 2$ by $x^{2} + 1$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{x^{2} + 3}{x^{2} + 1} = 2$ by $x^{2} + 1$ .

$\frac{x^{2} + 3}{x^{2} + 1} (x^{2} + 1) = 2 (x^{2} + 1)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $x^{2} + 1$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{x^{2} + 3}{x^{2} + 1} (x^{2} + 1) = 2 (x^{2} + 1)$

Step 2.2.1.2

Rewrite the expression.

$x^{2} + 3 = 2 (x^{2} + 1)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Apply the distributive property.

$x^{2} + 3 = 2 x^{2} + 2 \cdot 1$

Step 2.3.2

Multiply $2$ by $1$ .

$x^{2} + 3 = 2 x^{2} + 2$

Step 3

Solve the equation.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Subtract $2 x^{2}$ from $x^{2}$ .

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

Step 3.2

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

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Step 3.2.1

Subtract $3$ from both sides of the equation.

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

Step 3.2.2

Subtract $3$ from $2$ .

$- x^{2} = - 1$

Step 3.3

Divide each term in $- x^{2} = - 1$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- x^{2} = - 1$ by $- 1$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

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

Step 3.3.2.2

Divide $x^{2}$ by $1$ .

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 1$ by $- 1$ .

$x^{2} = 1$

Step 3.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1}$

Step 3.5

Any root of $1$ is $1$ .

$x = \pm 1$

Step 3.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Step 3.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

Step 3.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$