Algebra Examples

$\frac{x^{2}}{2 x - 6} = \frac{9}{6 x - 18}$

Step 1

Factor each term.

Tap for more steps...

Step 1.1

Factor $2$ out of $2 x - 6$ .

Tap for more steps...

Step 1.1.1

Factor $2$ out of $2 x$ .

$\frac{x^{2}}{2 (x) - 6} = \frac{9}{6 x - 18}$

Step 1.1.2

Factor $2$ out of $- 6$ .

$\frac{x^{2}}{2 x + 2 \cdot - 3} = \frac{9}{6 x - 18}$

Step 1.1.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

$\frac{x^{2}}{2 (x - 3)} = \frac{9}{6 x - 18}$

Step 1.2

Factor $6$ out of $6 x - 18$ .

Tap for more steps...

Step 1.2.1

Factor $6$ out of $6 x$ .

$\frac{x^{2}}{2 (x - 3)} = \frac{9}{6 (x) - 18}$

Step 1.2.2

Factor $6$ out of $- 18$ .

$\frac{x^{2}}{2 (x - 3)} = \frac{9}{6 x + 6 \cdot - 3}$

Step 1.2.3

Factor $6$ out of $6 x + 6 \cdot - 3$ .

$\frac{x^{2}}{2 (x - 3)} = \frac{9}{6 (x - 3)}$

Step 1.3

Reduce the expression $\frac{9}{6 (x - 3)}$ by cancelling the common factors.

Tap for more steps...

Step 1.3.1

Factor $3$ out of $9$ .

$\frac{x^{2}}{2 (x - 3)} = \frac{3 (3)}{6 (x - 3)}$

Step 1.3.2

Factor $3$ out of $6 (x - 3)$ .

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

Step 1.3.3

Cancel the common factor.

$\frac{x^{2}}{2 (x - 3)} = \frac{3 \cdot 3}{3 (2 (x - 3))}$

Step 1.3.4

Rewrite the expression.

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

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

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

$2 (x - 3), 2 (x - 3)$

Step 2.2

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 2.3

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 2.4

The LCM of $2, 2$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

Step 2.5

The factor for $x - 3$ is $x - 3$ itself.

$(x - 3) = x - 3$

$(x - 3)$ occurs $1$ time.

Step 2.6

The LCM of $x - 3, x - 3$ is the result of multiplying all factors the greatest number of times they occur in either term.

$x - 3$

Step 2.7

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$2 (x - 3)$

Step 3

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

Tap for more steps...

Step 3.1

Multiply each term in $\frac{x^{2}}{2 (x - 3)} = \frac{3}{2 (x - 3)}$ by $2 (x - 3)$ .

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

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Rewrite the expression.

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

Step 3.2.3

Cancel the common factor of $x - 3$ .

Tap for more steps...

Step 3.2.3.1

Cancel the common factor.

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

Step 3.2.3.2

Rewrite the expression.

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

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

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

Step 3.3.3

Cancel the common factor of $x - 3$ .

Tap for more steps...

Step 3.3.3.1

Cancel the common factor.

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

Step 3.3.3.2

Rewrite the expression.

$x^{2} = 3$

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

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

$x = \pm \sqrt{3}$

Step 4.2

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

Tap for more steps...

Step 4.2.1

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

$x = \sqrt{3}$

Step 4.2.2

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

$x = - \sqrt{3}$

Step 4.2.3

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

$x = \sqrt{3}, - \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{3}, - \sqrt{3}$

Decimal Form:

$x = 1.73205080 \dots, - 1.73205080 \dots$