Algebra Examples

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

Step 1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

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

Step 2

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

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Step 2.1

Subtract $2 x$ from both sides of the equation.

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

Step 2.2

Combine the opposite terms in $x^{2} + 2 x + 2 - 2 x$ .

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Step 2.2.1

Subtract $2 x$ from $2 x$ .

$x^{2} + 0 + 2 = 3$

Step 2.2.2

Add $x^{2}$ and $0$ .

$x^{2} + 2 = 3$

Step 3

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

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Step 3.1

Subtract $2$ from both sides of the equation.

$x^{2} = 3 - 2$

Step 3.2

Subtract $2$ from $3$ .

$x^{2} = 1$

Step 4

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

$x = \pm \sqrt{1}$

Step 5

Any root of $1$ is $1$ .

$x = \pm 1$

Step 6

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

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Step 6.1

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

$x = 1$

Step 6.2

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

$x = - 1$

Step 6.3

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

$x = 1, - 1$

Step 7

Exclude the solutions that do not make $\frac{x^{2} + 2 x + 2}{x - 1} = \frac{2 x + 3}{x - 1}$ true.

$x = - 1$