Algebra Examples

$x^{2} + \frac{2}{5} x + \frac{1}{25}$

Step 1

Rewrite the middle term.

$x^{2} + 2 x \frac{1}{5} + 0 + \frac{1}{25}$

Step 2

Rearrange terms.

$x^{2} + 2 x \frac{1}{5} + \frac{1}{25} + 0$

Step 3

Factor first three terms by perfect square rule.

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

Step 4

Rewrite ${(x + \frac{1}{5})}^{2}$ as $(x + \frac{1}{5}) (x + \frac{1}{5})$ .

$(x + \frac{1}{5}) (x + \frac{1}{5}) + 0$

Step 5

Expand $(x + \frac{1}{5}) (x + \frac{1}{5})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$x (x + \frac{1}{5}) + \frac{1}{5} (x + \frac{1}{5}) + 0$

Step 5.2

Apply the distributive property.

$x \cdot x + x \frac{1}{5} + \frac{1}{5} (x + \frac{1}{5}) + 0$

Step 5.3

Apply the distributive property.

$x \cdot x + x \frac{1}{5} + \frac{1}{5} x + \frac{1}{5} \cdot \frac{1}{5} + 0$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $x$ by $x$ .

$x^{2} + x \frac{1}{5} + \frac{1}{5} x + \frac{1}{5} \cdot \frac{1}{5} + 0$

Step 6.1.2

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

$x^{2} + \frac{x}{5} + \frac{1}{5} x + \frac{1}{5} \cdot \frac{1}{5} + 0$

Step 6.1.3

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

$x^{2} + \frac{x}{5} + \frac{x}{5} + \frac{1}{5} \cdot \frac{1}{5} + 0$

Step 6.1.4

Multiply $\frac{1}{5} \cdot \frac{1}{5}$ .

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Step 6.1.4.1

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

$x^{2} + \frac{x}{5} + \frac{x}{5} + \frac{1}{5 \cdot 5} + 0$

Step 6.1.4.2

Multiply $5$ by $5$ .

$x^{2} + \frac{x}{5} + \frac{x}{5} + \frac{1}{25} + 0$

Step 6.2

Add $\frac{x}{5}$ and $\frac{x}{5}$ .

$x^{2} + 2 \frac{x}{5} + \frac{1}{25} + 0$

Step 7

Combine $2$ and $\frac{x}{5}$ .

$x^{2} + \frac{2 x}{5} + \frac{1}{25} + 0$

Step 8

Add $x^{2} + \frac{2 x}{5} + \frac{1}{25}$ and $0$ .

$x^{2} + \frac{2 x}{5} + \frac{1}{25}$

Step 9

Factor using the perfect square rule.

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Step 9.1

Rewrite $1$ as $1^{2}$ .

$x^{2} + \frac{2 x}{5} + \frac{1^{2}}{25}$

Step 9.2

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

$x^{2} + \frac{2 x}{5} + \frac{1^{2}}{5^{2}}$

Step 9.3

Rewrite $\frac{1^{2}}{5^{2}}$ as ${(\frac{1}{5})}^{2}$ .

$x^{2} + \frac{2 x}{5} + {(\frac{1}{5})}^{2}$

Step 9.4

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

$\frac{2 x}{5} = 2 \cdot x \cdot \frac{1}{5}$

Step 9.5

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot \frac{1}{5} + {(\frac{1}{5})}^{2}$

Step 9.6

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = \frac{1}{5}$ .

${(x + \frac{1}{5})}^{2}$