Factor 1/4x^2+x+1

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

Step 1

Factor using the perfect square rule.

Rewrite $\frac{1}{4} x^{2}$ as ${(\frac{1}{2} x)}^{2}$ .

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

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

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

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

$x = 2 \cdot (\frac{1}{2} x) \cdot 1$

Rewrite the polynomial.

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

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

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

Step 2

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

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