Simplify (2x+3)/(x-4)-(x-4)/(2x+3)

$\frac{2 x + 3}{x - 4} - \frac{x - 4}{2 x + 3}$

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

$\frac{2 x + 3}{x - 4} \cdot \frac{2 x + 3}{2 x + 3} - \frac{x - 4}{2 x + 3}$

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

$\frac{2 x + 3}{x - 4} \cdot \frac{2 x + 3}{2 x + 3} - \frac{x - 4}{2 x + 3} \cdot \frac{x - 4}{x - 4}$

Write each expression with a common denominator of $(x - 4) (2 x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $(x - 4) (2 x + 3)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Raise $2 x + 3$ to the power of $1$ .

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

Raise $2 x + 3$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Rewrite $(x - 4) (x - 4)$ as ${(x - 4)}^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x + 3$ and $b = x - 4$ .

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

Simplify.

Add $2 x$ and $x$ .

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

Subtract $4$ from $3$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $- 4$ .

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

Subtract $x$ from $2 x$ .

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

Add $3$ and $4$ .

$\frac{(3 x - 1) (x + 7)}{(2 x + 3) (x - 4)}$

Simplify (2x+3)/(x-4)-(x-4)/(2x+3)

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