Simplify square root of 4x^2+2x+1/4

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

Factor using the perfect square rule.

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

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

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

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

Rewrite $4$ as $2^{2}$ .

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

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

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

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

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

Rewrite the polynomial.

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the expression.

Multiply $2$ by $2$ .

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

Apply the product rule to $\frac{4 x + 1}{2}$ .

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

Raise $2$ to the power of $2$ .

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

Rewrite $4$ as $2^{2}$ .

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

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

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

Pull terms out from under the radical, assuming positive real numbers.

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