Answer:
\(\frac{1}{9}\) is to be added to make \(x^2+\frac{2}{3}x\) a perfect square trinomial as \((x+\frac{1}{3})^{2}\)
Step-by-step explanation:
Given : \(x^2+\frac{2}{3}x\)
We have to find what can be added to \(x^2+\frac{2}{3}x\) to form a perfect square trinomial.
Perfect square trinomial is of the form \((a+b)^2\)
Consider the given expression \(x^2+\frac{2}{3}x\)
Using algebraic identity \((a+b)^2=a^2+b^2+2ab\) ,
Comparing , we get, a = x ......(1)
\(2ab=\frac{2}{3}x\)
Uisng (1) , we get,
\(2b=\frac{2}{3} \\\\ \Rightarrow b=\frac{1}{3}\)
To make it a perfect square trinomial we need to add \(b^2\) term
\(b^2=\frac{1}{9}\)
Thus, \(\frac{1}{9}\) is to be added to make \(x^2+\frac{2}{3}x\) a perfect square trinomial as \((x+\frac{1}{3})^{2}\)
