Solve by Factoring (4(x-2))/(x-3)+3/x=-3/(x(x-3))

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

Step 1

Subtract $\frac{- 3}{x (x - 3)}$ from both sides of the equation.

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

Step 2

Simplify $\frac{4 (x - 2)}{x - 3} + \frac{3}{x} - \frac{- 3}{x (x - 3)}$ .

Simplify each term.

Move the negative in front of the fraction.

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

Multiply $- - \frac{3}{x (x - 3)}$ .

Multiply $- 1$ by $- 1$ .

$\frac{4 (x - 2)}{x - 3} + \frac{3}{x} + 1 \frac{3}{x (x - 3)} = 0$

Multiply $\frac{3}{x (x - 3)}$ by $1$ .

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

To write $\frac{4 (x - 2)}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

To write $\frac{3}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

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

Write each expression with a common denominator of $(x - 3) x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4 (x - 2)}{x - 3}$ by $\frac{x}{x}$ .

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

Multiply $\frac{3}{x}$ by $\frac{x - 3}{x - 3}$ .

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

Reorder the factors of $(x - 3) x$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $4$ by $- 2$ .

$\frac{(4 x - 8) x + 3 (x - 3)}{x (x - 3)} + \frac{3}{x (x - 3)} = 0$

Apply the distributive property.

$\frac{4 x \cdot x - 8 x + 3 (x - 3)}{x (x - 3)} + \frac{3}{x (x - 3)} = 0$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{4 (x \cdot x) - 8 x + 3 (x - 3)}{x (x - 3)} + \frac{3}{x (x - 3)} = 0$

Multiply $x$ by $x$ .

$\frac{4 x^{2} - 8 x + 3 (x - 3)}{x (x - 3)} + \frac{3}{x (x - 3)} = 0$

Apply the distributive property.

$\frac{4 x^{2} - 8 x + 3 x + 3 \cdot - 3}{x (x - 3)} + \frac{3}{x (x - 3)} = 0$

Multiply $3$ by $- 3$ .

$\frac{4 x^{2} - 8 x + 3 x - 9}{x (x - 3)} + \frac{3}{x (x - 3)} = 0$

Add $- 8 x$ and $3 x$ .

$\frac{4 x^{2} - 5 x - 9}{x (x - 3)} + \frac{3}{x (x - 3)} = 0$

Factor by grouping.

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 $a \cdot c = 4 \cdot - 9 = - 36$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

$\frac{4 x^{2} - 5 (x) - 9}{x (x - 3)} + \frac{3}{x (x - 3)} = 0$

Rewrite $- 5$ as $4$ plus $- 9$

$\frac{4 x^{2} + (4 - 9) x - 9}{x (x - 3)} + \frac{3}{x (x - 3)} = 0$

Apply the distributive property.

$\frac{4 x^{2} + 4 x - 9 x - 9}{x (x - 3)} + \frac{3}{x (x - 3)} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(4 x^{2} + 4 x) - 9 x - 9}{x (x - 3)} + \frac{3}{x (x - 3)} = 0$

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

$\frac{4 x (x + 1) - 9 (x + 1)}{x (x - 3)} + \frac{3}{x (x - 3)} = 0$

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$\frac{(x + 1) (4 x - 9)}{x (x - 3)} + \frac{3}{x (x - 3)} = 0$

Combine the numerators over the common denominator.

$\frac{(x + 1) (4 x - 9) + 3}{x (x - 3)} = 0$

Simplify the numerator.

Expand $(x + 1) (4 x - 9)$ using the FOIL Method.

Apply the distributive property.

$\frac{x (4 x - 9) + 1 (4 x - 9) + 3}{x (x - 3)} = 0$

Apply the distributive property.

$\frac{x (4 x) + x \cdot - 9 + 1 (4 x - 9) + 3}{x (x - 3)} = 0$

Apply the distributive property.

$\frac{x (4 x) + x \cdot - 9 + 1 (4 x) + 1 \cdot - 9 + 3}{x (x - 3)} = 0$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$\frac{4 x \cdot x + x \cdot - 9 + 1 (4 x) + 1 \cdot - 9 + 3}{x (x - 3)} = 0$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{4 (x \cdot x) + x \cdot - 9 + 1 (4 x) + 1 \cdot - 9 + 3}{x (x - 3)} = 0$

Multiply $x$ by $x$ .

$\frac{4 x^{2} + x \cdot - 9 + 1 (4 x) + 1 \cdot - 9 + 3}{x (x - 3)} = 0$

Move $- 9$ to the left of $x$ .

$\frac{4 x^{2} - 9 \cdot x + 1 (4 x) + 1 \cdot - 9 + 3}{x (x - 3)} = 0$

Multiply $4 x$ by $1$ .

$\frac{4 x^{2} - 9 x + 4 x + 1 \cdot - 9 + 3}{x (x - 3)} = 0$

Multiply $- 9$ by $1$ .

$\frac{4 x^{2} - 9 x + 4 x - 9 + 3}{x (x - 3)} = 0$

Add $- 9 x$ and $4 x$ .

$\frac{4 x^{2} - 5 x - 9 + 3}{x (x - 3)} = 0$

Add $- 9$ and $3$ .

$\frac{4 x^{2} - 5 x - 6}{x (x - 3)} = 0$

Factor by grouping.

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 $a \cdot c = 4 \cdot - 6 = - 24$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

$\frac{4 x^{2} - 5 (x) - 6}{x (x - 3)} = 0$

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

$\frac{4 x^{2} + (3 - 8) x - 6}{x (x - 3)} = 0$

Apply the distributive property.

$\frac{4 x^{2} + 3 x - 8 x - 6}{x (x - 3)} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(4 x^{2} + 3 x) - 8 x - 6}{x (x - 3)} = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $4 x + 3$ .

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

Step 3

Set the numerator equal to zero.

$(4 x + 3) (x - 2) = 0$

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 $0$ .

$4 x + 3 = 0$

$x - 2 = 0$

Set $4 x + 3$ equal to $0$ and solve for $x$ .

Set $4 x + 3$ equal to $0$ .

$4 x + 3 = 0$

Solve $4 x + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$4 x = - 3$

Divide each term in $4 x = - 3$ by $4$ and simplify.

Divide each term in $4 x = - 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Divide $x$ by $1$ .

$x = \frac{- 3}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(4 x + 3) (x - 2) = 0$ true.

$x = - \frac{3}{4}, 2$

Solve by Factoring (4(x-2))/(x-3)+3/x=-3/(x(x-3))

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