Pre-Algebra Examples

\frac{4}{3 x^{2} + 15 x} + \frac{7 x}{x^{2} + 10 x + 25}

$\frac{4}{3 x^{2} + 15 x} + \frac{7 x}{x^{2} + 10 x + 25}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Factor

3 x

$3 x$ out of

3 x^{2} + 15 x

$3 x^{2} + 15 x$ .

Tap for more steps...

Step 1.1.1

Factor

3 x

$3 x$ out of

3 x^{2}

$3 x^{2}$ .

\frac{4}{3 x (x) + 15 x} + \frac{7 x}{x^{2} + 10 x + 25}

$\frac{4}{3 x (x) + 15 x} + \frac{7 x}{x^{2} + 10 x + 25}$

Step 1.1.2

Factor

3 x

$3 x$ out of

15 x

$15 x$ .

\frac{4}{3 x (x) + 3 x (5)} + \frac{7 x}{x^{2} + 10 x + 25}

$\frac{4}{3 x (x) + 3 x (5)} + \frac{7 x}{x^{2} + 10 x + 25}$

Step 1.1.3

Factor

3 x

$3 x$ out of

3 x (x) + 3 x (5)

$3 x (x) + 3 x (5)$ .

\frac{4}{3 x (x + 5)} + \frac{7 x}{x^{2} + 10 x + 25}

$\frac{4}{3 x (x + 5)} + \frac{7 x}{x^{2} + 10 x + 25}$

\frac{4}{3 x (x + 5)} + \frac{7 x}{x^{2} + 10 x + 25}

$\frac{4}{3 x (x + 5)} + \frac{7 x}{x^{2} + 10 x + 25}$

Step 1.2

Factor using the perfect square rule.

Tap for more steps...

Step 1.2.1

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

\frac{4}{3 x (x + 5)} + \frac{7 x}{x^{2} + 10 x + 5^{2}}

$\frac{4}{3 x (x + 5)} + \frac{7 x}{x^{2} + 10 x + 5^{2}}$

Step 1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

10 x = 2 \cdot x \cdot 5

$10 x = 2 \cdot x \cdot 5$

Step 1.2.3

Rewrite the polynomial.

\frac{4}{3 x (x + 5)} + \frac{7 x}{x^{2} + 2 \cdot x \cdot 5 + 5^{2}}

$\frac{4}{3 x (x + 5)} + \frac{7 x}{x^{2} + 2 \cdot x \cdot 5 + 5^{2}}$

Step 1.2.4

Factor using the perfect square trinomial rule

a^{2} + 2 a b + b^{2} = {(a + b)}^{2}

$a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where

a = x

$a = x$ and

b = 5

$b = 5$ .

\frac{4}{3 x (x + 5)} + \frac{7 x}{{(x + 5)}^{2}}

$\frac{4}{3 x (x + 5)} + \frac{7 x}{{(x + 5)}^{2}}$

\frac{4}{3 x (x + 5)} + \frac{7 x}{{(x + 5)}^{2}}

$\frac{4}{3 x (x + 5)} + \frac{7 x}{{(x + 5)}^{2}}$

\frac{4}{3 x (x + 5)} + \frac{7 x}{{(x + 5)}^{2}}

$\frac{4}{3 x (x + 5)} + \frac{7 x}{{(x + 5)}^{2}}$

Step 2

To write

\frac{4}{3 x (x + 5)}

$\frac{4}{3 x (x + 5)}$ as a fraction with a common denominator, multiply by

\frac{x + 5}{x + 5}

$\frac{x + 5}{x + 5}$ .

\frac{4}{3 x (x + 5)} \cdot \frac{x + 5}{x + 5} + \frac{7 x}{{(x + 5)}^{2}}

$\frac{4}{3 x (x + 5)} \cdot \frac{x + 5}{x + 5} + \frac{7 x}{{(x + 5)}^{2}}$

Step 3

To write

\frac{7 x}{{(x + 5)}^{2}}

$\frac{7 x}{{(x + 5)}^{2}}$ as a fraction with a common denominator, multiply by

\frac{3 x}{3 x}

$\frac{3 x}{3 x}$ .

\frac{4}{3 x (x + 5)} \cdot \frac{x + 5}{x + 5} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}

$\frac{4}{3 x (x + 5)} \cdot \frac{x + 5}{x + 5} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}$

Step 4

Write each expression with a common denominator of

3 x {(x + 5)}^{2}

$3 x {(x + 5)}^{2}$ , by multiplying each by an appropriate factor of

1

$1$ .

Tap for more steps...

Step 4.1

Multiply

\frac{4}{3 x (x + 5)}

$\frac{4}{3 x (x + 5)}$ by

\frac{x + 5}{x + 5}

$\frac{x + 5}{x + 5}$ .

\frac{4 (x + 5)}{3 x (x + 5) (x + 5)} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}

$\frac{4 (x + 5)}{3 x (x + 5) (x + 5)} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}$

Step 4.2

Raise

x + 5

$x + 5$ to the power of

1

$1$ .

\frac{4 (x + 5)}{3 x ({(x + 5)}^{1} (x + 5))} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}

$\frac{4 (x + 5)}{3 x ({(x + 5)}^{1} (x + 5))} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}$

Step 4.3

Raise

x + 5

$x + 5$ to the power of

1

$1$ .

\frac{4 (x + 5)}{3 x ({(x + 5)}^{1} {(x + 5)}^{1})} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}

$\frac{4 (x + 5)}{3 x ({(x + 5)}^{1} {(x + 5)}^{1})} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}$

Step 4.4

Use the power rule

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

$\frac{4 (x + 5)}{3 x {(x + 5)}^{1 + 1}} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}$

Step 4.5

Add

$1$ and

$1$ .

$\frac{4 (x + 5)}{3 x {(x + 5)}^{2}} + \frac{7 x}{{(x + 5)}^{2}} \cdot \frac{3 x}{3 x}$

Step 4.6

Multiply

$\frac{7 x}{{(x + 5)}^{2}}$ by

$\frac{3 x}{3 x}$ .

$\frac{4 (x + 5)}{3 x {(x + 5)}^{2}} + \frac{7 x (3 x)}{{(x + 5)}^{2} (3 x)}$

Step 4.7

Reorder the factors of

${(x + 5)}^{2} (3 x)$ .

$\frac{4 (x + 5)}{3 x {(x + 5)}^{2}} + \frac{7 x (3 x)}{3 x {(x + 5)}^{2}}$

Step 5

Combine the numerators over the common denominator.

$\frac{4 (x + 5) + 7 x (3 x)}{3 x {(x + 5)}^{2}}$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Apply the distributive property.

$\frac{4 x + 4 \cdot 5 + 7 x (3 x)}{3 x {(x + 5)}^{2}}$

Step 6.2

Multiply

$4$ by

$5$ .

$\frac{4 x + 20 + 7 x (3 x)}{3 x {(x + 5)}^{2}}$

Step 6.3

Rewrite using the commutative property of multiplication.

$\frac{4 x + 20 + 7 \cdot 3 x \cdot x}{3 x {(x + 5)}^{2}}$

Step 6.4

Multiply

$x$ by

$x$ by adding the exponents.

Tap for more steps...

Step 6.4.1

Move

$x$ .

$\frac{4 x + 20 + 7 \cdot 3 (x \cdot x)}{3 x {(x + 5)}^{2}}$

Step 6.4.2

Multiply

$x$ by

$x$ .

$\frac{4 x + 20 + 7 \cdot 3 x^{2}}{3 x {(x + 5)}^{2}}$

Step 6.5

Multiply

$7$ by

$3$ .

$\frac{4 x + 20 + 21 x^{2}}{3 x {(x + 5)}^{2}}$

Step 6.6

Reorder terms.

$\frac{21 x^{2} + 4 x + 20}{3 x {(x + 5)}^{2}}$