Basic Math Examples

$\frac{1}{25} k^{2} + \frac{6}{5} k + 9$

Step 1

Rewrite the middle term.

$\frac{1}{25} k^{2} + 2 (\frac{1}{5} k) \cdot 3 + 0 + 9$

Step 2

Rearrange terms.

$\frac{1}{25} k^{2} + 2 (\frac{1}{5} k) \cdot 3 + 9 + 0$

Step 3

Factor first three terms by perfect square rule.

${(\frac{1}{5} k + 3)}^{2} + 0$

Step 4

Combine $\frac{1}{5}$ and $k$ .

${(\frac{k}{5} + 3)}^{2} + 0$

Step 5

Rewrite ${(\frac{k}{5} + 3)}^{2}$ as $(\frac{k}{5} + 3) (\frac{k}{5} + 3)$ .

$(\frac{k}{5} + 3) (\frac{k}{5} + 3) + 0$

Step 6

Expand $(\frac{k}{5} + 3) (\frac{k}{5} + 3)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$\frac{k}{5} (\frac{k}{5} + 3) + 3 (\frac{k}{5} + 3) + 0$

Step 6.2

Apply the distributive property.

$\frac{k}{5} \cdot \frac{k}{5} + \frac{k}{5} \cdot 3 + 3 (\frac{k}{5} + 3) + 0$

Step 6.3

Apply the distributive property.

$\frac{k}{5} \cdot \frac{k}{5} + \frac{k}{5} \cdot 3 + 3 \frac{k}{5} + 3 \cdot 3 + 0$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $\frac{k}{5} \cdot \frac{k}{5}$ .

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Step 7.1.1.1

Multiply $\frac{k}{5}$ by $\frac{k}{5}$ .

$\frac{k \cdot k}{5 \cdot 5} + \frac{k}{5} \cdot 3 + 3 \frac{k}{5} + 3 \cdot 3 + 0$

Step 7.1.1.2

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

$\frac{k^{1} k}{5 \cdot 5} + \frac{k}{5} \cdot 3 + 3 \frac{k}{5} + 3 \cdot 3 + 0$

Step 7.1.1.3

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

$\frac{k^{1} k^{1}}{5 \cdot 5} + \frac{k}{5} \cdot 3 + 3 \frac{k}{5} + 3 \cdot 3 + 0$

Step 7.1.1.4

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

$\frac{k^{1 + 1}}{5 \cdot 5} + \frac{k}{5} \cdot 3 + 3 \frac{k}{5} + 3 \cdot 3 + 0$

Step 7.1.1.5

Add $1$ and $1$ .

$\frac{k^{2}}{5 \cdot 5} + \frac{k}{5} \cdot 3 + 3 \frac{k}{5} + 3 \cdot 3 + 0$

Step 7.1.1.6

Multiply $5$ by $5$ .

$\frac{k^{2}}{25} + \frac{k}{5} \cdot 3 + 3 \frac{k}{5} + 3 \cdot 3 + 0$

Step 7.1.2

Combine $\frac{k}{5}$ and $3$ .

$\frac{k^{2}}{25} + \frac{k \cdot 3}{5} + 3 \frac{k}{5} + 3 \cdot 3 + 0$

Step 7.1.3

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

$\frac{k^{2}}{25} + \frac{3 \cdot k}{5} + 3 \frac{k}{5} + 3 \cdot 3 + 0$

Step 7.1.4

Multiply $3$ by $k$ .

$\frac{k^{2}}{25} + \frac{3 k}{5} + 3 \frac{k}{5} + 3 \cdot 3 + 0$

Step 7.1.5

Combine $3$ and $\frac{k}{5}$ .

$\frac{k^{2}}{25} + \frac{3 k}{5} + \frac{3 k}{5} + 3 \cdot 3 + 0$

Step 7.1.6

Multiply $3$ by $3$ .

$\frac{k^{2}}{25} + \frac{3 k}{5} + \frac{3 k}{5} + 9 + 0$

Step 7.2

Add $\frac{3 k}{5}$ and $\frac{3 k}{5}$ .

$\frac{k^{2}}{25} + 2 \frac{3 k}{5} + 9 + 0$

Step 8

Multiply $2 \frac{3 k}{5}$ .

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Step 8.1

Combine $2$ and $\frac{3 k}{5}$ .

$\frac{k^{2}}{25} + \frac{2 (3 k)}{5} + 9 + 0$

Step 8.2

Multiply $3$ by $2$ .

$\frac{k^{2}}{25} + \frac{6 k}{5} + 9 + 0$

Step 9

Add $\frac{k^{2}}{25} + \frac{6 k}{5} + 9$ and $0$ .

$\frac{k^{2}}{25} + \frac{6 k}{5} + 9$

Step 10

Factor using the perfect square rule.

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Step 10.1

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

$\frac{k^{2}}{5^{2}} + \frac{6 k}{5} + 9$

Step 10.2

Rewrite $\frac{k^{2}}{5^{2}}$ as ${(\frac{k}{5})}^{2}$ .

${(\frac{k}{5})}^{2} + \frac{6 k}{5} + 9$

Step 10.3

Rewrite $9$ as $3^{2}$ .

${(\frac{k}{5})}^{2} + \frac{6 k}{5} + 3^{2}$

Step 10.4

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

$\frac{6 k}{5} = 2 \cdot \frac{k}{5} \cdot 3$

Step 10.5

Rewrite the polynomial.

${(\frac{k}{5})}^{2} + 2 \cdot \frac{k}{5} \cdot 3 + 3^{2}$

Step 10.6

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = \frac{k}{5}$ and $b = 3$ .

${(\frac{k}{5} + 3)}^{2}$