Solve by Factoring (2x)/(x-2)=5+(4x^2)/(x-2)

$\frac{2 x}{x - 2} = 5 + \frac{4 x^{2}}{x - 2}$

Step 1

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

Subtract $5$ from both sides of the equation.

$\frac{2 x}{x - 2} - 5 = \frac{4 x^{2}}{x - 2}$

Subtract $\frac{4 x^{2}}{x - 2}$ from both sides of the equation.

$\frac{2 x}{x - 2} - 5 - \frac{4 x^{2}}{x - 2} = 0$

Step 2

Simplify $\frac{2 x}{x - 2} - 5 - \frac{4 x^{2}}{x - 2}$ .

Find the common denominator.

Write $- 5$ as a fraction with denominator $1$ .

$\frac{2 x}{x - 2} + \frac{- 5}{1} - \frac{4 x^{2}}{x - 2} = 0$

Multiply $\frac{- 5}{1}$ by $\frac{x - 2}{x - 2}$ .

$\frac{2 x}{x - 2} + \frac{- 5}{1} \cdot \frac{x - 2}{x - 2} - \frac{4 x^{2}}{x - 2} = 0$

Multiply $\frac{- 5}{1}$ by $\frac{x - 2}{x - 2}$ .

$\frac{2 x}{x - 2} + \frac{- 5 (x - 2)}{x - 2} - \frac{4 x^{2}}{x - 2} = 0$

Combine the numerators over the common denominator.

$\frac{2 x - 5 (x - 2) - 4 x^{2}}{x - 2} = 0$

Simplify each term.

Apply the distributive property.

$\frac{2 x - 5 x - 5 \cdot - 2 - 4 x^{2}}{x - 2} = 0$

Multiply $- 5$ by $- 2$ .

$\frac{2 x - 5 x + 10 - 4 x^{2}}{x - 2} = 0$

Subtract $5 x$ from $2 x$ .

$\frac{- 3 x + 10 - 4 x^{2}}{x - 2} = 0$

Factor by grouping.

Reorder terms.

$\frac{- 4 x^{2} - 3 x + 10}{x - 2} = 0$

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 $a \cdot c = - 4 \cdot 10 = - 40$ and whose sum is $b = - 3$ .

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

$\frac{- 4 x^{2} - 3 (x) + 10}{x - 2} = 0$

Rewrite $- 3$ as $5$ plus $- 8$

$\frac{- 4 x^{2} + (5 - 8) x + 10}{x - 2} = 0$

Apply the distributive property.

$\frac{- 4 x^{2} + 5 x - 8 x + 10}{x - 2} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- 4 x^{2} + 5 x) - 8 x + 10}{x - 2} = 0$

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

$\frac{x (- 4 x + 5) + 2 (- 4 x + 5)}{x - 2} = 0$

Factor the polynomial by factoring out the greatest common factor, $- 4 x + 5$ .

$\frac{(- 4 x + 5) (x + 2)}{x - 2} = 0$

Factor $- 1$ out of $- 4 x$ .

$\frac{(- (4 x) + 5) (x + 2)}{x - 2} = 0$

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{(- (4 x) - 1 (- 5)) (x + 2)}{x - 2} = 0$

Factor $- 1$ out of $- (4 x) - 1 (- 5)$ .

$\frac{- (4 x - 5) (x + 2)}{x - 2} = 0$

Rewrite $- (4 x - 5)$ as $- 1 (4 x - 5)$ .

$\frac{- 1 (4 x - 5) (x + 2)}{x - 2} = 0$

Move the negative in front of the fraction.

$- \frac{(4 x - 5) (x + 2)}{x - 2} = 0$

Step 3

Set the numerator equal to zero.

$(4 x - 5) (x + 2) = 0$

Step 4

Solve the equation for $x$ .

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

$4 x - 5 = 0$

$x + 2 = 0$

Set $4 x - 5$ equal to $0$ and solve for $x$ .

Set $4 x - 5$ equal to $0$ .

$4 x - 5 = 0$

Solve $4 x - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$4 x = 5$

Divide each term in $4 x = 5$ by $4$ and simplify.

Divide each term in $4 x = 5$ by $4$ .

$\frac{4 x}{4} = \frac{5}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{5}{4}$

Divide $x$ by $1$ .

$x = \frac{5}{4}$

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $(4 x - 5) (x + 2) = 0$ true.

$x = \frac{5}{4}, - 2$

Solve by Factoring (2x)/(x-2)=5+(4x^2)/(x-2)

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