Basic Math Examples

$\frac{5 x - 3}{7 x + 3} + \frac{8 x^{2} + 3}{7 x + 1}$

Step 1

To write $\frac{5 x - 3}{7 x + 3}$ as a fraction with a common denominator, multiply by $\frac{7 x + 1}{7 x + 1}$ .

$\frac{5 x - 3}{7 x + 3} \cdot \frac{7 x + 1}{7 x + 1} + \frac{8 x^{2} + 3}{7 x + 1}$

Step 2

To write $\frac{8 x^{2} + 3}{7 x + 1}$ as a fraction with a common denominator, multiply by $\frac{7 x + 3}{7 x + 3}$ .

$\frac{5 x - 3}{7 x + 3} \cdot \frac{7 x + 1}{7 x + 1} + \frac{8 x^{2} + 3}{7 x + 1} \cdot \frac{7 x + 3}{7 x + 3}$

Step 3

Write each expression with a common denominator of $(7 x + 3) (7 x + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5 x - 3}{7 x + 3}$ by $\frac{7 x + 1}{7 x + 1}$ .

$\frac{(5 x - 3) (7 x + 1)}{(7 x + 3) (7 x + 1)} + \frac{8 x^{2} + 3}{7 x + 1} \cdot \frac{7 x + 3}{7 x + 3}$

Step 3.2

Multiply $\frac{8 x^{2} + 3}{7 x + 1}$ by $\frac{7 x + 3}{7 x + 3}$ .

$\frac{(5 x - 3) (7 x + 1)}{(7 x + 3) (7 x + 1)} + \frac{(8 x^{2} + 3) (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 3.3

Reorder the factors of $(7 x + 3) (7 x + 1)$ .

$\frac{(5 x - 3) (7 x + 1)}{(7 x + 1) (7 x + 3)} + \frac{(8 x^{2} + 3) (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(5 x - 3) (7 x + 1) + (8 x^{2} + 3) (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(5 x - 3) (7 x + 1)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{5 x (7 x + 1) - 3 (7 x + 1) + (8 x^{2} + 3) (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 5.1.2

Apply the distributive property.

$\frac{5 x (7 x) + 5 x \cdot 1 - 3 (7 x + 1) + (8 x^{2} + 3) (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 5.1.3

Apply the distributive property.

$\frac{5 x (7 x) + 5 x \cdot 1 - 3 (7 x) - 3 \cdot 1 + (8 x^{2} + 3) (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{5 \cdot 7 x \cdot x + 5 x \cdot 1 - 3 (7 x) - 3 \cdot 1 + (8 x^{2} + 3) (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 5.2.1.2

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

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Step 5.2.1.2.1

Move $x$ .

$\frac{5 \cdot 7 (x \cdot x) + 5 x \cdot 1 - 3 (7 x) - 3 \cdot 1 + (8 x^{2} + 3) (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 5.2.1.2.2

Multiply $x$ by $x$ .

$\frac{5 \cdot 7 x^{2} + 5 x \cdot 1 - 3 (7 x) - 3 \cdot 1 + (8 x^{2} + 3) (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 5.2.1.3

Multiply $5$ by $7$ .

$\frac{35 x^{2} + 5 x \cdot 1 - 3 (7 x) - 3 \cdot 1 + (8 x^{2} + 3) (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 5.2.1.4

Multiply $5$ by $1$ .

$\frac{35 x^{2} + 5 x - 3 (7 x) - 3 \cdot 1 + (8 x^{2} + 3) (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 5.2.1.5

Multiply $7$ by $- 3$ .

$\frac{35 x^{2} + 5 x - 21 x - 3 \cdot 1 + (8 x^{2} + 3) (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 5.2.1.6

Multiply $- 3$ by $1$ .

$\frac{35 x^{2} + 5 x - 21 x - 3 + (8 x^{2} + 3) (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 5.2.2

Subtract $21 x$ from $5 x$ .

$\frac{35 x^{2} - 16 x - 3 + (8 x^{2} + 3) (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 5.3

Expand $(8 x^{2} + 3) (7 x + 3)$ using the FOIL Method.

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Step 5.3.1

Apply the distributive property.

$\frac{35 x^{2} - 16 x - 3 + 8 x^{2} (7 x + 3) + 3 (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 5.3.2

Apply the distributive property.

$\frac{35 x^{2} - 16 x - 3 + 8 x^{2} (7 x) + 8 x^{2} \cdot 3 + 3 (7 x + 3)}{(7 x + 1) (7 x + 3)}$

Step 5.3.3

Apply the distributive property.

$\frac{35 x^{2} - 16 x - 3 + 8 x^{2} (7 x) + 8 x^{2} \cdot 3 + 3 (7 x) + 3 \cdot 3}{(7 x + 1) (7 x + 3)}$

Step 5.4

Simplify each term.

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Step 5.4.1

Rewrite using the commutative property of multiplication.

$\frac{35 x^{2} - 16 x - 3 + 8 \cdot 7 x^{2} x + 8 x^{2} \cdot 3 + 3 (7 x) + 3 \cdot 3}{(7 x + 1) (7 x + 3)}$

Step 5.4.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 5.4.2.1

Move $x$ .

$\frac{35 x^{2} - 16 x - 3 + 8 \cdot 7 (x \cdot x^{2}) + 8 x^{2} \cdot 3 + 3 (7 x) + 3 \cdot 3}{(7 x + 1) (7 x + 3)}$

Step 5.4.2.2

Multiply $x$ by $x^{2}$ .

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Step 5.4.2.2.1

Raise $x$ to the power of $1$ .

$\frac{35 x^{2} - 16 x - 3 + 8 \cdot 7 (x^{1} x^{2}) + 8 x^{2} \cdot 3 + 3 (7 x) + 3 \cdot 3}{(7 x + 1) (7 x + 3)}$

Step 5.4.2.2.2

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

$\frac{35 x^{2} - 16 x - 3 + 8 \cdot 7 x^{1 + 2} + 8 x^{2} \cdot 3 + 3 (7 x) + 3 \cdot 3}{(7 x + 1) (7 x + 3)}$

Step 5.4.2.3

Add $1$ and $2$ .

$\frac{35 x^{2} - 16 x - 3 + 8 \cdot 7 x^{3} + 8 x^{2} \cdot 3 + 3 (7 x) + 3 \cdot 3}{(7 x + 1) (7 x + 3)}$

Step 5.4.3

Multiply $8$ by $7$ .

$\frac{35 x^{2} - 16 x - 3 + 56 x^{3} + 8 x^{2} \cdot 3 + 3 (7 x) + 3 \cdot 3}{(7 x + 1) (7 x + 3)}$

Step 5.4.4

Multiply $3$ by $8$ .

$\frac{35 x^{2} - 16 x - 3 + 56 x^{3} + 24 x^{2} + 3 (7 x) + 3 \cdot 3}{(7 x + 1) (7 x + 3)}$

Step 5.4.5

Multiply $7$ by $3$ .

$\frac{35 x^{2} - 16 x - 3 + 56 x^{3} + 24 x^{2} + 21 x + 3 \cdot 3}{(7 x + 1) (7 x + 3)}$

Step 5.4.6

Multiply $3$ by $3$ .

$\frac{35 x^{2} - 16 x - 3 + 56 x^{3} + 24 x^{2} + 21 x + 9}{(7 x + 1) (7 x + 3)}$

Step 5.5

Add $35 x^{2}$ and $24 x^{2}$ .

$\frac{59 x^{2} - 16 x - 3 + 56 x^{3} + 21 x + 9}{(7 x + 1) (7 x + 3)}$

Step 5.6

Add $- 16 x$ and $21 x$ .

$\frac{59 x^{2} + 5 x - 3 + 56 x^{3} + 9}{(7 x + 1) (7 x + 3)}$

Step 5.7

Add $- 3$ and $9$ .

$\frac{59 x^{2} + 5 x + 56 x^{3} + 6}{(7 x + 1) (7 x + 3)}$

Step 5.8

Reorder terms.

$\frac{56 x^{3} + 59 x^{2} + 5 x + 6}{(7 x + 1) (7 x + 3)}$