Basic Math Examples

$\frac{8 a + 5}{a^{2} \cdot (2 a) - 35} - \frac{6 a - 5}{a^{2} - 10 a + 25}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $25$ as $5^{2}$ .

$\frac{8 a + 5}{a^{2} \cdot (2 a) - 35} - \frac{6 a - 5}{a^{2} - 10 a + 5^{2}}$

Step 1.2

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

$10 a = 2 \cdot a \cdot 5$

Step 1.3

Rewrite the polynomial.

$\frac{8 a + 5}{a^{2} \cdot (2 a) - 35} - \frac{6 a - 5}{a^{2} - 2 \cdot a \cdot 5 + 5^{2}}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = a$ and $b = 5$ .

$\frac{8 a + 5}{a^{2} \cdot (2 a) - 35} - \frac{6 a - 5}{{(a - 5)}^{2}}$

Step 2

To write $\frac{8 a + 5}{a^{2} \cdot (2 a) - 35}$ as a fraction with a common denominator, multiply by $\frac{{(a - 5)}^{2}}{{(a - 5)}^{2}}$ .

$\frac{8 a + 5}{a^{2} \cdot (2 a) - 35} \cdot \frac{{(a - 5)}^{2}}{{(a - 5)}^{2}} - \frac{6 a - 5}{{(a - 5)}^{2}}$

Step 3

To write $- \frac{6 a - 5}{{(a - 5)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{a^{2} \cdot (2 a) - 35}{a^{2} \cdot (2 a) - 35}$ .

$\frac{8 a + 5}{a^{2} \cdot (2 a) - 35} \cdot \frac{{(a - 5)}^{2}}{{(a - 5)}^{2}} - \frac{6 a - 5}{{(a - 5)}^{2}} \cdot \frac{a^{2} \cdot (2 a) - 35}{a^{2} \cdot (2 a) - 35}$

Step 4

Write each expression with a common denominator of $(a^{2} \cdot (2 a) - 35) {(a - 5)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{8 a + 5}{a^{2} \cdot (2 a) - 35}$ by $\frac{{(a - 5)}^{2}}{{(a - 5)}^{2}}$ .

$\frac{(8 a + 5) {(a - 5)}^{2}}{(a^{2} \cdot (2 a) - 35) {(a - 5)}^{2}} - \frac{6 a - 5}{{(a - 5)}^{2}} \cdot \frac{a^{2} \cdot (2 a) - 35}{a^{2} \cdot (2 a) - 35}$

Step 4.2

Multiply $\frac{6 a - 5}{{(a - 5)}^{2}}$ by $\frac{a^{2} \cdot (2 a) - 35}{a^{2} \cdot (2 a) - 35}$ .

$\frac{(8 a + 5) {(a - 5)}^{2}}{(a^{2} \cdot (2 a) - 35) {(a - 5)}^{2}} - \frac{(6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 4.3

Reorder the factors of $(a^{2} \cdot (2 a) - 35) {(a - 5)}^{2}$ .

$\frac{(8 a + 5) {(a - 5)}^{2}}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)} - \frac{(6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(8 a + 5) {(a - 5)}^{2} - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6

Rewrite $\frac{(8 a + 5) {(a - 5)}^{2} - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$ in a factored form.

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Step 6.1

Rewrite ${(a - 5)}^{2}$ as $(a - 5) (a - 5)$ .

$\frac{(8 a + 5) ((a - 5) (a - 5)) - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.2

Expand $(a - 5) (a - 5)$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

$\frac{(8 a + 5) (a (a - 5) - 5 (a - 5)) - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.2.2

Apply the distributive property.

$\frac{(8 a + 5) (a \cdot a + a \cdot - 5 - 5 (a - 5)) - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.2.3

Apply the distributive property.

$\frac{(8 a + 5) (a \cdot a + a \cdot - 5 - 5 a - 5 \cdot - 5) - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.3

Simplify and combine like terms.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply $a$ by $a$ .

$\frac{(8 a + 5) (a^{2} + a \cdot - 5 - 5 a - 5 \cdot - 5) - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.3.1.2

Move $- 5$ to the left of $a$ .

$\frac{(8 a + 5) (a^{2} - 5 a - 5 a - 5 \cdot - 5) - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.3.1.3

Multiply $- 5$ by $- 5$ .

$\frac{(8 a + 5) (a^{2} - 5 a - 5 a + 25) - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.3.2

Subtract $5 a$ from $- 5 a$ .

$\frac{(8 a + 5) (a^{2} - 10 a + 25) - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.4

Expand $(8 a + 5) (a^{2} - 10 a + 25)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{8 a \cdot a^{2} + 8 a (- 10 a) + 8 a \cdot 25 + 5 a^{2} + 5 (- 10 a) + 5 \cdot 25 - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.5

Simplify each term.

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Step 6.5.1

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

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Step 6.5.1.1

Move $a^{2}$ .

$\frac{8 (a^{2} a) + 8 a (- 10 a) + 8 a \cdot 25 + 5 a^{2} + 5 (- 10 a) + 5 \cdot 25 - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.5.1.2

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

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Step 6.5.1.2.1

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

$\frac{8 (a^{2} a^{1}) + 8 a (- 10 a) + 8 a \cdot 25 + 5 a^{2} + 5 (- 10 a) + 5 \cdot 25 - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.5.1.2.2

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

$\frac{8 a^{2 + 1} + 8 a (- 10 a) + 8 a \cdot 25 + 5 a^{2} + 5 (- 10 a) + 5 \cdot 25 - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.5.1.3

Add $2$ and $1$ .

$\frac{8 a^{3} + 8 a (- 10 a) + 8 a \cdot 25 + 5 a^{2} + 5 (- 10 a) + 5 \cdot 25 - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.5.2

Rewrite using the commutative property of multiplication.

$\frac{8 a^{3} + 8 \cdot - 10 a \cdot a + 8 a \cdot 25 + 5 a^{2} + 5 (- 10 a) + 5 \cdot 25 - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.5.3

Multiply $a$ by $a$ by adding the exponents.

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Step 6.5.3.1

Move $a$ .

$\frac{8 a^{3} + 8 \cdot - 10 (a \cdot a) + 8 a \cdot 25 + 5 a^{2} + 5 (- 10 a) + 5 \cdot 25 - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.5.3.2

Multiply $a$ by $a$ .

$\frac{8 a^{3} + 8 \cdot - 10 a^{2} + 8 a \cdot 25 + 5 a^{2} + 5 (- 10 a) + 5 \cdot 25 - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.5.4

Multiply $8$ by $- 10$ .

$\frac{8 a^{3} - 80 a^{2} + 8 a \cdot 25 + 5 a^{2} + 5 (- 10 a) + 5 \cdot 25 - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.5.5

Multiply $25$ by $8$ .

$\frac{8 a^{3} - 80 a^{2} + 200 a + 5 a^{2} + 5 (- 10 a) + 5 \cdot 25 - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.5.6

Multiply $- 10$ by $5$ .

$\frac{8 a^{3} - 80 a^{2} + 200 a + 5 a^{2} - 50 a + 5 \cdot 25 - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.5.7

Multiply $5$ by $25$ .

$\frac{8 a^{3} - 80 a^{2} + 200 a + 5 a^{2} - 50 a + 125 - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.6

Add $- 80 a^{2}$ and $5 a^{2}$ .

$\frac{8 a^{3} - 75 a^{2} + 200 a - 50 a + 125 - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.7

Subtract $50 a$ from $200 a$ .

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 - (6 a - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.8

Apply the distributive property.

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 + (- (6 a) - - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.9

Multiply $6$ by $- 1$ .

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 + (- 6 a - - 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.10

Multiply $- 1$ by $- 5$ .

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 + (- 6 a + 5) (a^{2} \cdot (2 a) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.11

Simplify each term.

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Step 6.11.1

Rewrite using the commutative property of multiplication.

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 + (- 6 a + 5) (2 a^{2} a - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.11.2

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

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Step 6.11.2.1

Move $a$ .

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 + (- 6 a + 5) (2 (a \cdot a^{2}) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.11.2.2

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

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Step 6.11.2.2.1

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

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 + (- 6 a + 5) (2 (a^{1} a^{2}) - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.11.2.2.2

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

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 + (- 6 a + 5) (2 a^{1 + 2} - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.11.2.3

Add $1$ and $2$ .

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 + (- 6 a + 5) (2 a^{3} - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.12

Expand $(- 6 a + 5) (2 a^{3} - 35)$ using the FOIL Method.

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Step 6.12.1

Apply the distributive property.

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 - 6 a (2 a^{3} - 35) + 5 (2 a^{3} - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.12.2

Apply the distributive property.

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 - 6 a (2 a^{3}) - 6 a \cdot - 35 + 5 (2 a^{3} - 35)}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.12.3

Apply the distributive property.

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 - 6 a (2 a^{3}) - 6 a \cdot - 35 + 5 (2 a^{3}) + 5 \cdot - 35}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.13

Simplify each term.

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Step 6.13.1

Rewrite using the commutative property of multiplication.

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 - 6 \cdot 2 a \cdot a^{3} - 6 a \cdot - 35 + 5 (2 a^{3}) + 5 \cdot - 35}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.13.2

Multiply $a$ by $a^{3}$ by adding the exponents.

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Step 6.13.2.1

Move $a^{3}$ .

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 - 6 \cdot 2 (a^{3} a) - 6 a \cdot - 35 + 5 (2 a^{3}) + 5 \cdot - 35}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.13.2.2

Multiply $a^{3}$ by $a$ .

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Step 6.13.2.2.1

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

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 - 6 \cdot 2 (a^{3} a^{1}) - 6 a \cdot - 35 + 5 (2 a^{3}) + 5 \cdot - 35}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.13.2.2.2

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

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 - 6 \cdot 2 a^{3 + 1} - 6 a \cdot - 35 + 5 (2 a^{3}) + 5 \cdot - 35}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.13.2.3

Add $3$ and $1$ .

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 - 6 \cdot 2 a^{4} - 6 a \cdot - 35 + 5 (2 a^{3}) + 5 \cdot - 35}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.13.3

Multiply $- 6$ by $2$ .

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 - 12 a^{4} - 6 a \cdot - 35 + 5 (2 a^{3}) + 5 \cdot - 35}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.13.4

Multiply $- 35$ by $- 6$ .

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 - 12 a^{4} + 210 a + 5 (2 a^{3}) + 5 \cdot - 35}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.13.5

Multiply $2$ by $5$ .

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 - 12 a^{4} + 210 a + 10 a^{3} + 5 \cdot - 35}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.13.6

Multiply $5$ by $- 35$ .

$\frac{8 a^{3} - 75 a^{2} + 150 a + 125 - 12 a^{4} + 210 a + 10 a^{3} - 175}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.14

Add $8 a^{3}$ and $10 a^{3}$ .

$\frac{18 a^{3} - 75 a^{2} + 150 a + 125 - 12 a^{4} + 210 a - 175}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.15

Add $150 a$ and $210 a$ .

$\frac{18 a^{3} - 75 a^{2} + 360 a + 125 - 12 a^{4} - 175}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.16

Subtract $175$ from $125$ .

$\frac{18 a^{3} - 75 a^{2} + 360 a - 12 a^{4} - 50}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.17

Reorder terms.

$\frac{- 12 a^{4} + 18 a^{3} - 75 a^{2} + 360 a - 50}{{(a - 5)}^{2} (a^{2} \cdot (2 a) - 35)}$

Step 6.18

Rewrite using the commutative property of multiplication.

$\frac{- 12 a^{4} + 18 a^{3} - 75 a^{2} + 360 a - 50}{{(a - 5)}^{2} (2 a^{2} a - 35)}$

Step 6.19

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

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Step 6.19.1

Move $a$ .

$\frac{- 12 a^{4} + 18 a^{3} - 75 a^{2} + 360 a - 50}{{(a - 5)}^{2} (2 (a \cdot a^{2}) - 35)}$

Step 6.19.2

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

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Step 6.19.2.1

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

$\frac{- 12 a^{4} + 18 a^{3} - 75 a^{2} + 360 a - 50}{{(a - 5)}^{2} (2 (a^{1} a^{2}) - 35)}$

Step 6.19.2.2

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

$\frac{- 12 a^{4} + 18 a^{3} - 75 a^{2} + 360 a - 50}{{(a - 5)}^{2} (2 a^{1 + 2} - 35)}$

Step 6.19.3

Add $1$ and $2$ .

$\frac{- 12 a^{4} + 18 a^{3} - 75 a^{2} + 360 a - 50}{{(a - 5)}^{2} (2 a^{3} - 35)}$