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

`(1)/(9)` is to be added to make
`x^2+(2)/(3)x` a perfect square trinomial as
`(x+(1)/(3))^(2)`

Explanation:

Given :
`x^2+(2)/(3)x`

We have to find what can be added to
`x^2+(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+(2)/(3)x`

Using algebraic identity
`(a+b)^2=a^2+b^2+2ab` ,

Comparing , we get, a = x ......(1)

`2ab=(2)/(3)x`

Uisng (1) , we get,

`2b=(2)/(3) \\\\ \Rightarrow b=(1)/(3)`

To make it a perfect square trinomial we need to add
`b^2` term

`b^2=(1)/(9)`

Thus,
`(1)/(9)` is to be added to make
`x^2+(2)/(3)x` a perfect square trinomial as
`(x+(1)/(3))^(2)`