Algebra Examples

\frac{3 x}{2 y + 3} + \frac{x^{2} + 3 x}{4 x y - 3 - 2 y + 6 x}

$\frac{3 x}{2 y + 3} + \frac{x^{2} + 3 x}{4 x y - 3 - 2 y + 6 x}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Factor

x

$x$ out of

x^{2} + 3 x

$x^{2} + 3 x$ .

Tap for more steps...

Step 1.1.1

Factor

x

$x$ out of

x^{2}

$x^{2}$ .

\frac{3 x}{2 y + 3} + \frac{x \cdot x + 3 x}{4 x y - 3 - 2 y + 6 x}

$\frac{3 x}{2 y + 3} + \frac{x \cdot x + 3 x}{4 x y - 3 - 2 y + 6 x}$

Step 1.1.2

Factor

x

$x$ out of

3 x

$3 x$ .

\frac{3 x}{2 y + 3} + \frac{x \cdot x + x \cdot 3}{4 x y - 3 - 2 y + 6 x}

$\frac{3 x}{2 y + 3} + \frac{x \cdot x + x \cdot 3}{4 x y - 3 - 2 y + 6 x}$

Step 1.1.3

Factor

x

$x$ out of

x \cdot x + x \cdot 3

$x \cdot x + x \cdot 3$ .

\frac{3 x}{2 y + 3} + \frac{x (x + 3)}{4 x y - 3 - 2 y + 6 x}

$\frac{3 x}{2 y + 3} + \frac{x (x + 3)}{4 x y - 3 - 2 y + 6 x}$

\frac{3 x}{2 y + 3} + \frac{x (x + 3)}{4 x y - 3 - 2 y + 6 x}

$\frac{3 x}{2 y + 3} + \frac{x (x + 3)}{4 x y - 3 - 2 y + 6 x}$

Step 1.2

Simplify the denominator.

Tap for more steps...

Step 1.2.1

Reorder terms.

\frac{3 x}{2 y + 3} + \frac{x (x + 3)}{4 x y + 6 x - 2 y - 3}

$\frac{3 x}{2 y + 3} + \frac{x (x + 3)}{4 x y + 6 x - 2 y - 3}$

Step 1.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.2.1

Group the first two terms and the last two terms.

\frac{3 x}{2 y + 3} + \frac{x (x + 3)}{(4 x y + 6 x) - 2 y - 3}

$\frac{3 x}{2 y + 3} + \frac{x (x + 3)}{(4 x y + 6 x) - 2 y - 3}$

Step 1.2.2.2

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

\frac{3 x}{2 y + 3} + \frac{x (x + 3)}{2 x (2 y + 3) - (2 y + 3)}

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

\frac{3 x}{2 y + 3} + \frac{x (x + 3)}{2 x (2 y + 3) - (2 y + 3)}

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

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor,

2 y + 3

$2 y + 3$ .

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

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

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

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

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

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

Step 2

To write

\frac{3 x}{2 y + 3}

$\frac{3 x}{2 y + 3}$ as a fraction with a common denominator, multiply by

\frac{2 x - 1}{2 x - 1}

$\frac{2 x - 1}{2 x - 1}$ .

\frac{3 x}{2 y + 3} \cdot \frac{2 x - 1}{2 x - 1} + \frac{x (x + 3)}{(2 y + 3) (2 x - 1)}

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

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Multiply

\frac{3 x}{2 y + 3}

$\frac{3 x}{2 y + 3}$ by

\frac{2 x - 1}{2 x - 1}

$\frac{2 x - 1}{2 x - 1}$ .

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

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

Step 3.2

Combine the numerators over the common denominator.

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

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

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

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

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Factor

x

$x$ out of

3 x (2 x - 1) + x (x + 3)

$3 x (2 x - 1) + x (x + 3)$ .

Tap for more steps...

Step 4.1.1

Factor

x

$x$ out of

3 x (2 x - 1)

$3 x (2 x - 1)$ .

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

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

Step 4.1.2

Factor

x

$x$ out of

x (3 (2 x - 1)) + x (x + 3)

$x (3 (2 x - 1)) + x (x + 3)$ .

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

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

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

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

Step 4.2

Apply the distributive property.

\frac{x (3 (2 x) + 3 \cdot - 1 + x + 3)}{(2 y + 3) (2 x - 1)}

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

Step 4.3

Multiply

2

$2$ by

3

$3$ .

\frac{x (6 x + 3 \cdot - 1 + x + 3)}{(2 y + 3) (2 x - 1)}

$\frac{x (6 x + 3 \cdot - 1 + x + 3)}{(2 y + 3) (2 x - 1)}$

Step 4.4

Multiply

3

$3$ by

- 1

$- 1$ .

\frac{x (6 x - 3 + x + 3)}{(2 y + 3) (2 x - 1)}

$\frac{x (6 x - 3 + x + 3)}{(2 y + 3) (2 x - 1)}$

Step 4.5

Add

6 x

$6 x$ and

x

$x$ .

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

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

Step 4.6

Add

- 3

$- 3$ and

3

$3$ .

\frac{x (7 x + 0)}{(2 y + 3) (2 x - 1)}

$\frac{x (7 x + 0)}{(2 y + 3) (2 x - 1)}$

Step 4.7

Add

7 x

$7 x$ and

0

$0$ .

\frac{x \cdot 7 x}{(2 y + 3) (2 x - 1)}

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

Step 4.8

Combine exponents.

Tap for more steps...

Step 4.8.1

Raise

x

$x$ to the power of

1

$1$ .

\frac{7 (x^{1} x)}{(2 y + 3) (2 x - 1)}

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

Step 4.8.2

Raise

x

$x$ to the power of

1

$1$ .

\frac{7 (x^{1} x^{1})}{(2 y + 3) (2 x - 1)}

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

Step 4.8.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{7 x^{1 + 1}}{(2 y + 3) (2 x - 1)}

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

Step 4.8.4

Add

1

$1$ and

1

$1$ .

\frac{7 x^{2}}{(2 y + 3) (2 x - 1)}

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

\frac{7 x^{2}}{(2 y + 3) (2 x - 1)}

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

\frac{7 x^{2}}{(2 y + 3) (2 x - 1)}

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