which value must be added to the expression $x^{2}+x$ to make it a perfect - square trinomial?\n$\frac{1}{4}$…

Explanation:

Step1: Recall the formula for a perfect - square trinomial

The general form of a perfect - square trinomial is $(a + b)^2=a^{2}+2ab + b^{2}$. In the given expression $x^{2}+x$, we have $a = x$ and $2ab=x$. Since $a = x$, then $2xb=x$.

Step2: Solve for $b$

From $2xb=x$, divide both sides by $2x$ (assuming $x\neq0$; when $x = 0$, the concept still holds). We get $b=\frac{1}{2}$.

Step3: Find the value to add

We need to add $b^{2}$ to the expression $x^{2}+2ab$ to make it a perfect - square trinomial. Since $b=\frac{1}{2}$, then $b^{2}=\left(\frac{1}{2}\right)^{2}=\frac{1}{4}$.

Answer:

A. $\frac{1}{4}$