Question

Which of the following constants can be added to x^2 + 2/3x to form a perfect square trinomial?

Answer 1

take 1/2 of the linear coefient and square it

2/3 times 1/2=1/3
square 1/3 to get 1/9

the constnat is 1/9
it would factor to (x+1/3)^2

Answer 2

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 $(x+\frac{1}{3})^{2}$