Factor x^2+1/4x+1/64

$x^{2} + \frac{1}{4} x + \frac{1}{64}$

Step 1

Rewrite the middle term.

$x^{2} + 2 x \frac{1}{8} + 0 + \frac{1}{64}$

Step 2

Rearrange terms.

$x^{2} + 2 x \frac{1}{8} + \frac{1}{64} + 0$

Step 3

Factor first three terms by perfect square rule.

${(x + \frac{1}{8})}^{2} + 0$

Step 4

Rewrite ${(x + \frac{1}{8})}^{2}$ as $(x + \frac{1}{8}) (x + \frac{1}{8})$ .

$(x + \frac{1}{8}) (x + \frac{1}{8}) + 0$

Step 5

Expand $(x + \frac{1}{8}) (x + \frac{1}{8})$ using the FOIL Method.

Apply the distributive property.

$x (x + \frac{1}{8}) + \frac{1}{8} (x + \frac{1}{8}) + 0$

Apply the distributive property.

$x \cdot x + x \frac{1}{8} + \frac{1}{8} (x + \frac{1}{8}) + 0$

Apply the distributive property.

$x \cdot x + x \frac{1}{8} + \frac{1}{8} x + \frac{1}{8} \cdot \frac{1}{8} + 0$

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \frac{1}{8} + \frac{1}{8} x + \frac{1}{8} \cdot \frac{1}{8} + 0$

Combine $x$ and $\frac{1}{8}$ .

$x^{2} + \frac{x}{8} + \frac{1}{8} x + \frac{1}{8} \cdot \frac{1}{8} + 0$

Combine $\frac{1}{8}$ and $x$ .

$x^{2} + \frac{x}{8} + \frac{x}{8} + \frac{1}{8} \cdot \frac{1}{8} + 0$

Multiply $\frac{1}{8} \cdot \frac{1}{8}$ .

Multiply $\frac{1}{8}$ by $\frac{1}{8}$ .

$x^{2} + \frac{x}{8} + \frac{x}{8} + \frac{1}{8 \cdot 8} + 0$

Multiply $8$ by $8$ .

$x^{2} + \frac{x}{8} + \frac{x}{8} + \frac{1}{64} + 0$

Add $\frac{x}{8}$ and $\frac{x}{8}$ .

$x^{2} + 2 \frac{x}{8} + \frac{1}{64} + 0$

Step 7

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

$x^{2} + 2 \frac{x}{2 (4)} + \frac{1}{64} + 0$

Cancel the common factor.

$x^{2} + 2 \frac{x}{2 \cdot 4} + \frac{1}{64} + 0$

Rewrite the expression.

$x^{2} + \frac{x}{4} + \frac{1}{64} + 0$

Step 8

Add $x^{2} + \frac{x}{4} + \frac{1}{64}$ and $0$ .

$x^{2} + \frac{x}{4} + \frac{1}{64}$

Step 9

Factor using the perfect square rule.

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

$x^{2} + \frac{x}{4} + \frac{1^{2}}{64}$

Rewrite $64$ as $8^{2}$ .

$x^{2} + \frac{x}{4} + \frac{1^{2}}{8^{2}}$

Rewrite $\frac{1^{2}}{8^{2}}$ as ${(\frac{1}{8})}^{2}$ .

$x^{2} + \frac{x}{4} + {(\frac{1}{8})}^{2}$

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

$\frac{x}{4} = 2 \cdot x \cdot \frac{1}{8}$

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot \frac{1}{8} + {(\frac{1}{8})}^{2}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = \frac{1}{8}$ .

${(x + \frac{1}{8})}^{2}$

Factor x^2+1/4x+1/64

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