Algebra Examples

$\frac{6 x}{x - 6} = 9 + \frac{7 x^{2}}{x - 6}$

Step 1

Move all the expressions to the left side of the equation.

Tap for more steps...

Step 1.1

Subtract $9$ from both sides of the equation.

$\frac{6 x}{x - 6} - 9 = \frac{7 x^{2}}{x - 6}$

Step 1.2

Subtract $\frac{7 x^{2}}{x - 6}$ from both sides of the equation.

$\frac{6 x}{x - 6} - 9 - \frac{7 x^{2}}{x - 6} = 0$

Step 2

Simplify $\frac{6 x}{x - 6} - 9 - \frac{7 x^{2}}{x - 6}$ .

Tap for more steps...

Step 2.1

Find the common denominator.

Tap for more steps...

Step 2.1.1

Write $- 9$ as a fraction with denominator $1$ .

$\frac{6 x}{x - 6} + \frac{- 9}{1} - \frac{7 x^{2}}{x - 6} = 0$

Step 2.1.2

Multiply $\frac{- 9}{1}$ by $\frac{x - 6}{x - 6}$ .

$\frac{6 x}{x - 6} + \frac{- 9}{1} \cdot \frac{x - 6}{x - 6} - \frac{7 x^{2}}{x - 6} = 0$

Step 2.1.3

Multiply $\frac{- 9}{1}$ by $\frac{x - 6}{x - 6}$ .

$\frac{6 x}{x - 6} + \frac{- 9 (x - 6)}{x - 6} - \frac{7 x^{2}}{x - 6} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{6 x - 9 (x - 6) - 7 x^{2}}{x - 6} = 0$

Step 2.3

Simplify each term.

Tap for more steps...

Step 2.3.1

Apply the distributive property.

$\frac{6 x - 9 x - 9 \cdot - 6 - 7 x^{2}}{x - 6} = 0$

Step 2.3.2

Multiply $- 9$ by $- 6$ .

$\frac{6 x - 9 x + 54 - 7 x^{2}}{x - 6} = 0$

Step 2.4

Subtract $9 x$ from $6 x$ .

$\frac{- 3 x + 54 - 7 x^{2}}{x - 6} = 0$

Step 2.5

Factor by grouping.

Tap for more steps...

Step 2.5.1

Reorder terms.

$\frac{- 7 x^{2} - 3 x + 54}{x - 6} = 0$

Step 2.5.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 7 \cdot 54 = - 378$ and whose sum is $b = - 3$ .

Tap for more steps...

Step 2.5.2.1

Factor $- 3$ out of $- 3 x$ .

$\frac{- 7 x^{2} - 3 (x) + 54}{x - 6} = 0$

Step 2.5.2.2

Rewrite $- 3$ as $18$ plus $- 21$

$\frac{- 7 x^{2} + (18 - 21) x + 54}{x - 6} = 0$

Step 2.5.2.3

Apply the distributive property.

$\frac{- 7 x^{2} + 18 x - 21 x + 54}{x - 6} = 0$

Step 2.5.3

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.5.3.1

Group the first two terms and the last two terms.

$\frac{(- 7 x^{2} + 18 x) - 21 x + 54}{x - 6} = 0$

Step 2.5.3.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (- 7 x + 18) + 3 (- 7 x + 18)}{x - 6} = 0$

Step 2.5.4

Factor the polynomial by factoring out the greatest common factor, $- 7 x + 18$ .

$\frac{(- 7 x + 18) (x + 3)}{x - 6} = 0$

Step 2.6

Factor $- 1$ out of $- 7 x$ .

$\frac{(- (7 x) + 18) (x + 3)}{x - 6} = 0$

Step 2.7

Rewrite $18$ as $- 1 (- 18)$ .

$\frac{(- (7 x) - 1 (- 18)) (x + 3)}{x - 6} = 0$

Step 2.8

Factor $- 1$ out of $- (7 x) - 1 (- 18)$ .

$\frac{- (7 x - 18) (x + 3)}{x - 6} = 0$

Step 2.9

Rewrite $- (7 x - 18)$ as $- 1 (7 x - 18)$ .

$\frac{- 1 (7 x - 18) (x + 3)}{x - 6} = 0$

Step 2.10

Move the negative in front of the fraction.

$- \frac{(7 x - 18) (x + 3)}{x - 6} = 0$

Step 3

Set the numerator equal to zero.

$(7 x - 18) (x + 3) = 0$

Step 4

Solve the equation for $x$ .

Tap for more steps...

Step 4.1

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

$7 x - 18 = 0$

$x + 3 = 0$

Step 4.2

Set $7 x - 18$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.2.1

Set $7 x - 18$ equal to $0$ .

$7 x - 18 = 0$

Step 4.2.2

Solve $7 x - 18 = 0$ for $x$ .

Tap for more steps...

Step 4.2.2.1

Add $18$ to both sides of the equation.

$7 x = 18$

Step 4.2.2.2

Divide each term in $7 x = 18$ by $7$ and simplify.

Tap for more steps...

Step 4.2.2.2.1

Divide each term in $7 x = 18$ by $7$ .

$\frac{7 x}{7} = \frac{18}{7}$

Step 4.2.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.2.2.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.2.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{18}{7}$

Step 4.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{18}{7}$

Step 4.3

Set $x + 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.3.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 4.3.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4.4

The final solution is all the values that make $(7 x - 18) (x + 3) = 0$ true.

$x = \frac{18}{7}, - 3$