Algebra Examples

$\frac{1}{2} g^{2} + \frac{7}{2} + 3 g^{2} - \frac{4}{5} g + \frac{1}{4}$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{g^{2}}{2} + \frac{7}{2} + 3 g^{2} - \frac{4}{5} g + \frac{1}{4}$

Step 1.2

Combine $g$ and $\frac{4}{5}$ .

$\frac{g^{2}}{2} + \frac{7}{2} + 3 g^{2} - \frac{g \cdot 4}{5} + \frac{1}{4}$

Step 1.3

Move $4$ to the left of $g$ .

$\frac{g^{2}}{2} + \frac{7}{2} + 3 g^{2} - \frac{4 g}{5} + \frac{1}{4}$

Step 2

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

$\frac{g^{2}}{2} + 3 g^{2} \cdot \frac{2}{2} + \frac{7}{2} - \frac{4 g}{5} + \frac{1}{4}$

Step 3

Simplify terms.

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Step 3.1

Combine $3 g^{2}$ and $\frac{2}{2}$ .

$\frac{g^{2}}{2} + \frac{3 g^{2} \cdot 2}{2} + \frac{7}{2} - \frac{4 g}{5} + \frac{1}{4}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{g^{2} + 3 g^{2} \cdot 2}{2} + \frac{7}{2} - \frac{4 g}{5} + \frac{1}{4}$

Step 3.3

Combine the numerators over the common denominator.

$\frac{g^{2} + 3 g^{2} \cdot 2 + 7}{2} + \frac{- 4 g}{5} + \frac{1}{4}$

Step 3.4

Multiply $2$ by $3$ .

$\frac{g^{2} + 6 g^{2} + 7}{2} + \frac{- 4 g}{5} + \frac{1}{4}$

Step 3.5

Add $g^{2}$ and $6 g^{2}$ .

$\frac{7 g^{2} + 7}{2} + \frac{- 4 g}{5} + \frac{1}{4}$

Step 4

Simplify each term.

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Step 4.1

Factor $7$ out of $7 g^{2} + 7$ .

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Step 4.1.1

Factor $7$ out of $7 g^{2}$ .

$\frac{7 (g^{2}) + 7}{2} + \frac{- 4 g}{5} + \frac{1}{4}$

Step 4.1.2

Factor $7$ out of $7$ .

$\frac{7 (g^{2}) + 7 (1)}{2} + \frac{- 4 g}{5} + \frac{1}{4}$

Step 4.1.3

Factor $7$ out of $7 (g^{2}) + 7 (1)$ .

$\frac{7 (g^{2} + 1)}{2} + \frac{- 4 g}{5} + \frac{1}{4}$

Step 4.2

Move the negative in front of the fraction.

$\frac{7 (g^{2} + 1)}{2} - \frac{4 g}{5} + \frac{1}{4}$

Step 5

To write $\frac{7 (g^{2} + 1)}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{7 (g^{2} + 1)}{2} \cdot \frac{5}{5} - \frac{4 g}{5} + \frac{1}{4}$

Step 6

To write $- \frac{4 g}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7 (g^{2} + 1)}{2} \cdot \frac{5}{5} - \frac{4 g}{5} \cdot \frac{2}{2} + \frac{1}{4}$

Step 7

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{7 (g^{2} + 1)}{2}$ by $\frac{5}{5}$ .

$\frac{7 (g^{2} + 1) \cdot 5}{2 \cdot 5} - \frac{4 g}{5} \cdot \frac{2}{2} + \frac{1}{4}$

Step 7.2

Multiply $2$ by $5$ .

$\frac{7 (g^{2} + 1) \cdot 5}{10} - \frac{4 g}{5} \cdot \frac{2}{2} + \frac{1}{4}$

Step 7.3

Multiply $\frac{4 g}{5}$ by $\frac{2}{2}$ .

$\frac{7 (g^{2} + 1) \cdot 5}{10} - \frac{4 g \cdot 2}{5 \cdot 2} + \frac{1}{4}$

Step 7.4

Multiply $5$ by $2$ .

$\frac{7 (g^{2} + 1) \cdot 5}{10} - \frac{4 g \cdot 2}{10} + \frac{1}{4}$

Step 8

Combine the numerators over the common denominator.

$\frac{7 (g^{2} + 1) \cdot 5 - 4 g \cdot 2}{10} + \frac{1}{4}$

Step 9

Simplify the numerator.

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Step 9.1

Apply the distributive property.

$\frac{(7 g^{2} + 7 \cdot 1) \cdot 5 - 4 g \cdot 2}{10} + \frac{1}{4}$

Step 9.2

Multiply $7$ by $1$ .

$\frac{(7 g^{2} + 7) \cdot 5 - 4 g \cdot 2}{10} + \frac{1}{4}$

Step 9.3

Apply the distributive property.

$\frac{7 g^{2} \cdot 5 + 7 \cdot 5 - 4 g \cdot 2}{10} + \frac{1}{4}$

Step 9.4

Multiply $5$ by $7$ .

$\frac{35 g^{2} + 7 \cdot 5 - 4 g \cdot 2}{10} + \frac{1}{4}$

Step 9.5

Multiply $7$ by $5$ .

$\frac{35 g^{2} + 35 - 4 g \cdot 2}{10} + \frac{1}{4}$

Step 9.6

Multiply $2$ by $- 4$ .

$\frac{35 g^{2} + 35 - 8 g}{10} + \frac{1}{4}$

Step 9.7

Reorder terms.

$\frac{35 g^{2} - 8 g + 35}{10} + \frac{1}{4}$

Step 10

To write $\frac{35 g^{2} - 8 g + 35}{10}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{35 g^{2} - 8 g + 35}{10} \cdot \frac{2}{2} + \frac{1}{4}$

Step 11

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{35 g^{2} - 8 g + 35}{10} \cdot \frac{2}{2} + \frac{1}{4} \cdot \frac{5}{5}$

Step 12

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.1

Multiply $\frac{35 g^{2} - 8 g + 35}{10}$ by $\frac{2}{2}$ .

$\frac{(35 g^{2} - 8 g + 35) \cdot 2}{10 \cdot 2} + \frac{1}{4} \cdot \frac{5}{5}$

Step 12.2

Multiply $10$ by $2$ .

$\frac{(35 g^{2} - 8 g + 35) \cdot 2}{20} + \frac{1}{4} \cdot \frac{5}{5}$

Step 12.3

Multiply $\frac{1}{4}$ by $\frac{5}{5}$ .

$\frac{(35 g^{2} - 8 g + 35) \cdot 2}{20} + \frac{5}{4 \cdot 5}$

Step 12.4

Multiply $4$ by $5$ .

$\frac{(35 g^{2} - 8 g + 35) \cdot 2}{20} + \frac{5}{20}$

Step 13

Combine the numerators over the common denominator.

$\frac{(35 g^{2} - 8 g + 35) \cdot 2 + 5}{20}$

Step 14

Simplify the numerator.

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Step 14.1

Apply the distributive property.

$\frac{35 g^{2} \cdot 2 - 8 g \cdot 2 + 35 \cdot 2 + 5}{20}$

Step 14.2

Simplify.

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Step 14.2.1

Multiply $2$ by $35$ .

$\frac{70 g^{2} - 8 g \cdot 2 + 35 \cdot 2 + 5}{20}$

Step 14.2.2

Multiply $2$ by $- 8$ .

$\frac{70 g^{2} - 16 g + 35 \cdot 2 + 5}{20}$

Step 14.2.3

Multiply $35$ by $2$ .

$\frac{70 g^{2} - 16 g + 70 + 5}{20}$

Step 14.3

Add $70$ and $5$ .

$\frac{70 g^{2} - 16 g + 75}{20}$