Basic Math Examples

$\frac{a^{2} - 3 a + 2}{25 a} - \frac{a - 1}{5 a^{2}}$

Step 1

Factor $a^{2} - 3 a + 2$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 1.2

Write the factored form using these integers.

$\frac{(a - 2) (a - 1)}{25 a} - \frac{a - 1}{5 a^{2}}$

Step 2

To write $\frac{(a - 2) (a - 1)}{25 a}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$\frac{(a - 2) (a - 1)}{25 a} \cdot \frac{a}{a} - \frac{a - 1}{5 a^{2}}$

Step 3

To write $- \frac{a - 1}{5 a^{2}}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{(a - 2) (a - 1)}{25 a} \cdot \frac{a}{a} - \frac{a - 1}{5 a^{2}} \cdot \frac{5}{5}$

Step 4

Write each expression with a common denominator of $25 a^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{(a - 2) (a - 1)}{25 a}$ by $\frac{a}{a}$ .

$\frac{(a - 2) (a - 1) a}{25 a \cdot a} - \frac{a - 1}{5 a^{2}} \cdot \frac{5}{5}$

Step 4.2

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

$\frac{(a - 2) (a - 1) a}{25 (a^{1} a)} - \frac{a - 1}{5 a^{2}} \cdot \frac{5}{5}$

Step 4.3

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

$\frac{(a - 2) (a - 1) a}{25 (a^{1} a^{1})} - \frac{a - 1}{5 a^{2}} \cdot \frac{5}{5}$

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

$\frac{(a - 2) (a - 1) a}{25 a^{2}} - \frac{a - 1}{5 a^{2}} \cdot \frac{5}{5}$

Step 4.6

Multiply $\frac{a - 1}{5 a^{2}}$ by $\frac{5}{5}$ .

$\frac{(a - 2) (a - 1) a}{25 a^{2}} - \frac{(a - 1) \cdot 5}{5 a^{2} \cdot 5}$

Step 4.7

Multiply $5$ by $5$ .

$\frac{(a - 2) (a - 1) a}{25 a^{2}} - \frac{(a - 1) \cdot 5}{25 a^{2}}$

Step 5

Combine the numerators over the common denominator.

$\frac{(a - 2) (a - 1) a - (a - 1) \cdot 5}{25 a^{2}}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $a - 1$ out of $(a - 2) (a - 1) a - (a - 1) \cdot 5$ .

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Step 6.1.1

Factor $a - 1$ out of $(a - 2) (a - 1) a$ .

$\frac{(a - 1) ((a - 2) a) - (a - 1) \cdot 5}{25 a^{2}}$

Step 6.1.2

Factor $a - 1$ out of $- (a - 1) \cdot 5$ .

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

Step 6.1.3

Factor $a - 1$ out of $(a - 1) ((a - 2) a) + (a - 1) (- 1 \cdot 5)$ .

$\frac{(a - 1) ((a - 2) a - 1 \cdot 5)}{25 a^{2}}$

Step 6.2

Apply the distributive property.

$\frac{(a - 1) (a \cdot a - 2 a - 1 \cdot 5)}{25 a^{2}}$

Step 6.3

Multiply $a$ by $a$ .

$\frac{(a - 1) (a^{2} - 2 a - 1 \cdot 5)}{25 a^{2}}$

Step 6.4

Multiply $- 1$ by $5$ .

$\frac{(a - 1) (a^{2} - 2 a - 5)}{25 a^{2}}$