Algebra Examples

$- \frac{5}{4} a^{2} - \frac{3}{2} a b + \frac{1}{2} a^{2} + 5 a b - 3 a^{2} - \frac{1}{2} a b$

Step 1

Simplify each term.

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Step 1.1

Combine $a^{2}$ and $\frac{5}{4}$ .

$- \frac{a^{2} \cdot 5}{4} - \frac{3}{2} \cdot (a b) + \frac{1}{2} a^{2} + 5 a b - 3 a^{2} - \frac{1}{2} \cdot (a b)$

Step 1.2

Move $5$ to the left of $a^{2}$ .

$- \frac{5 \cdot a^{2}}{4} - \frac{3}{2} \cdot (a b) + \frac{1}{2} a^{2} + 5 a b - 3 a^{2} - \frac{1}{2} \cdot (a b)$

Step 1.3

Multiply $- \frac{3}{2} (a b)$ .

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Step 1.3.1

Combine $a$ and $\frac{3}{2}$ .

$- \frac{5 a^{2}}{4} - \frac{a \cdot 3}{2} b + \frac{1}{2} a^{2} + 5 a b - 3 a^{2} - \frac{1}{2} \cdot (a b)$

Step 1.3.2

Combine $b$ and $\frac{a \cdot 3}{2}$ .

$- \frac{5 a^{2}}{4} - \frac{b (a \cdot 3)}{2} + \frac{1}{2} a^{2} + 5 a b - 3 a^{2} - \frac{1}{2} \cdot (a b)$

Step 1.4

Move $3$ to the left of $b a$ .

$- \frac{5 a^{2}}{4} - \frac{3 \cdot (b a)}{2} + \frac{1}{2} a^{2} + 5 a b - 3 a^{2} - \frac{1}{2} \cdot (a b)$

Step 1.5

Combine $\frac{1}{2}$ and $a^{2}$ .

$- \frac{5 a^{2}}{4} - \frac{3 b a}{2} + \frac{a^{2}}{2} + 5 a b - 3 a^{2} - \frac{1}{2} \cdot (a b)$

Step 1.6

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

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Step 1.6.1

Combine $a$ and $\frac{1}{2}$ .

$- \frac{5 a^{2}}{4} - \frac{3 b a}{2} + \frac{a^{2}}{2} + 5 a b - 3 a^{2} - \frac{a}{2} b$

Step 1.6.2

Combine $b$ and $\frac{a}{2}$ .

$- \frac{5 a^{2}}{4} - \frac{3 b a}{2} + \frac{a^{2}}{2} + 5 a b - 3 a^{2} - \frac{b a}{2}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$5 a b - 3 a^{2} + \frac{- 5 a^{2}}{4} + \frac{- 3 b a + a^{2} - b a}{2}$

Step 2.2

Subtract $b a$ from $- 3 b a$ .

$5 a b - 3 a^{2} + \frac{- 5 a^{2}}{4} + \frac{a^{2} - 4 b a}{2}$

Step 3

Simplify each term.

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Step 3.1

Move the negative in front of the fraction.

$5 a b - 3 a^{2} - \frac{5 a^{2}}{4} + \frac{a^{2} - 4 b a}{2}$

Step 3.2

Factor $a$ out of $a^{2} - 4 b a$ .

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Step 3.2.1

Factor $a$ out of $a^{2}$ .

$5 a b - 3 a^{2} - \frac{5 a^{2}}{4} + \frac{a \cdot a - 4 b a}{2}$

Step 3.2.2

Factor $a$ out of $- 4 b a$ .

$5 a b - 3 a^{2} - \frac{5 a^{2}}{4} + \frac{a \cdot a + a (- 4 b)}{2}$

Step 3.2.3

Factor $a$ out of $a \cdot a + a (- 4 b)$ .

$5 a b - 3 a^{2} - \frac{5 a^{2}}{4} + \frac{a (a - 4 b)}{2}$

Step 4

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

$- 3 a^{2} - \frac{5 a^{2}}{4} + 5 a b \cdot \frac{2}{2} + \frac{a (a - 4 b)}{2}$

Step 5

Simplify terms.

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Step 5.1

Combine $5 a b$ and $\frac{2}{2}$ .

$- 3 a^{2} - \frac{5 a^{2}}{4} + \frac{5 a b \cdot 2}{2} + \frac{a (a - 4 b)}{2}$

Step 5.2

Combine the numerators over the common denominator.

$- 3 a^{2} - \frac{5 a^{2}}{4} + \frac{5 a b \cdot 2 + a (a - 4 b)}{2}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $a$ out of $5 a b \cdot 2 + a (a - 4 b)$ .

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Step 6.1.1

Factor $a$ out of $5 a b \cdot 2$ .

$- 3 a^{2} - \frac{5 a^{2}}{4} + \frac{a (5 b \cdot 2) + a (a - 4 b)}{2}$

Step 6.1.2

Factor $a$ out of $a (5 b \cdot 2) + a (a - 4 b)$ .

$- 3 a^{2} - \frac{5 a^{2}}{4} + \frac{a (5 b \cdot 2 + a - 4 b)}{2}$

Step 6.2

Multiply $2$ by $5$ .

$- 3 a^{2} - \frac{5 a^{2}}{4} + \frac{a (10 b + a - 4 b)}{2}$

Step 6.3

Subtract $4 b$ from $10 b$ .

$- 3 a^{2} - \frac{5 a^{2}}{4} + \frac{a (a + 6 b)}{2}$

Step 7

To write $- 3 a^{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- 3 a^{2} \cdot \frac{4}{4} - \frac{5 a^{2}}{4} + \frac{a (a + 6 b)}{2}$

Step 8

Simplify terms.

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Step 8.1

Combine $- 3 a^{2}$ and $\frac{4}{4}$ .

$\frac{- 3 a^{2} \cdot 4}{4} - \frac{5 a^{2}}{4} + \frac{a (a + 6 b)}{2}$

Step 8.2

Combine the numerators over the common denominator.

$\frac{- 3 a^{2} \cdot 4 - 5 a^{2}}{4} + \frac{a (a + 6 b)}{2}$

Step 9

Simplify each term.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Factor $a^{2}$ out of $- 3 a^{2} \cdot 4 - 5 a^{2}$ .

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Step 9.1.1.1

Factor $a^{2}$ out of $- 3 a^{2} \cdot 4$ .

$\frac{a^{2} (- 3 \cdot 4) - 5 a^{2}}{4} + \frac{a (a + 6 b)}{2}$

Step 9.1.1.2

Factor $a^{2}$ out of $- 5 a^{2}$ .

$\frac{a^{2} (- 3 \cdot 4) + a^{2} \cdot - 5}{4} + \frac{a (a + 6 b)}{2}$

Step 9.1.1.3

Factor $a^{2}$ out of $a^{2} (- 3 \cdot 4) + a^{2} \cdot - 5$ .

$\frac{a^{2} (- 3 \cdot 4 - 5)}{4} + \frac{a (a + 6 b)}{2}$

Step 9.1.2

Multiply $- 3$ by $4$ .

$\frac{a^{2} (- 12 - 5)}{4} + \frac{a (a + 6 b)}{2}$

Step 9.1.3

Subtract $5$ from $- 12$ .

$\frac{a^{2} \cdot - 17}{4} + \frac{a (a + 6 b)}{2}$

Step 9.2

Move $- 17$ to the left of $a^{2}$ .

$\frac{- 17 \cdot a^{2}}{4} + \frac{a (a + 6 b)}{2}$

Step 9.3

Move the negative in front of the fraction.

$- \frac{17 a^{2}}{4} + \frac{a (a + 6 b)}{2}$

Step 10

To write $\frac{a (a + 6 b)}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{17 a^{2}}{4} + \frac{a (a + 6 b)}{2} \cdot \frac{2}{2}$

Step 11

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.1

Multiply $\frac{a (a + 6 b)}{2}$ by $\frac{2}{2}$ .

$- \frac{17 a^{2}}{4} + \frac{a (a + 6 b) \cdot 2}{2 \cdot 2}$

Step 11.2

Multiply $2$ by $2$ .

$- \frac{17 a^{2}}{4} + \frac{a (a + 6 b) \cdot 2}{4}$

Step 12

Combine the numerators over the common denominator.

$\frac{- 17 a^{2} + a (a + 6 b) \cdot 2}{4}$

Step 13

Simplify the numerator.

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Step 13.1

Factor $a$ out of $- 17 a^{2} + a (a + 6 b) \cdot 2$ .

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Step 13.1.1

Factor $a$ out of $- 17 a^{2}$ .

$\frac{a (- 17 a) + a (a + 6 b) \cdot 2}{4}$

Step 13.1.2

Factor $a$ out of $a (a + 6 b) \cdot 2$ .

$\frac{a (- 17 a) + a ((a + 6 b) \cdot 2)}{4}$

Step 13.1.3

Factor $a$ out of $a (- 17 a) + a ((a + 6 b) \cdot 2)$ .

$\frac{a (- 17 a + (a + 6 b) \cdot 2)}{4}$

Step 13.2

Apply the distributive property.

$\frac{a (- 17 a + a \cdot 2 + 6 b \cdot 2)}{4}$

Step 13.3

Move $2$ to the left of $a$ .

$\frac{a (- 17 a + 2 \cdot a + 6 b \cdot 2)}{4}$

Step 13.4

Multiply $2$ by $6$ .

$\frac{a (- 17 a + 2 \cdot a + 12 b)}{4}$

Step 13.5

Add $- 17 a$ and $2 a$ .

$\frac{a (- 15 a + 12 b)}{4}$

Step 13.6

Factor $3$ out of $- 15 a + 12 b$ .

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Step 13.6.1

Factor $3$ out of $- 15 a$ .

$\frac{a (3 (- 5 a) + 12 b)}{4}$

Step 13.6.2

Factor $3$ out of $12 b$ .

$\frac{a (3 (- 5 a) + 3 (4 b))}{4}$

Step 13.6.3

Factor $3$ out of $3 (- 5 a) + 3 (4 b)$ .

$\frac{a (3 (- 5 a + 4 b))}{4}$

$\frac{a \cdot 3 (- 5 a + 4 b)}{4}$

Step 14

Simplify with factoring out.

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Step 14.1

Move $3$ to the left of $a$ .

$\frac{3 \cdot a (- 5 a + 4 b)}{4}$

Step 14.2

Factor $- 1$ out of $- 5 a$ .

$\frac{3 a (- (5 a) + 4 b)}{4}$

Step 14.3

Factor $- 1$ out of $4 b$ .

$\frac{3 a (- (5 a) - (- 4 b))}{4}$

Step 14.4

Factor $- 1$ out of $- (5 a) - (- 4 b)$ .

$\frac{3 a (- (5 a - 4 b))}{4}$

Step 14.5

Simplify the expression.

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Step 14.5.1

Rewrite $- (5 a - 4 b)$ as $- 1 (5 a - 4 b)$ .

$\frac{3 a (- 1 (5 a - 4 b))}{4}$

Step 14.5.2

Move the negative in front of the fraction.

$- \frac{(3 a) (5 a - 4 b)}{4}$

Step 14.5.3

Reorder factors in $- \frac{(3 a) (5 a - 4 b)}{4}$ .

$- \frac{3 a (5 a - 4 b)}{4}$