Question

Please I'm in urgent need of help. what must be added to make (1/4)x^2-(2/3)xy a perfect square?.​

Answer 1

`\bigg((2)/(3) {y} \bigg)^(2)`

STEP BY STEP EXPLANATION

`(1)/(4) {x}^(2) - \bigg( (2)/(3) \bigg)xy \\ \\ = \bigg((1)/(2) {x} \bigg)^(2) - 2. \bigg((1)/(2) {x} \bigg)\bigg( (2)/(3) y \bigg ) + \bigg((2)/(3) {y} \bigg)^(2) \\ \\ = \bigg((1)/(2) {x} - (2)/(3) y\bigg)^(2) \\`

To make
`\red{\bold{(1)/(4) {x}^(2) - \bigg( (2)/(3) \bigg)xy}}` a perfect square we should add
`\purple{\bold{\bigg((2)/(3) {y} \bigg)^(2)}}`

Answer 2

• 
  `4/9y^2`

Explanation:

Given:

• 
  `(1/4)x^2-(2/3)xy`

Consider the identity:

• 
  `(a - b)^2=a^2-2ab+b^2`

We can see that:

• 
  `a^2=(1/2x)^2`
• 
  `a = (1/2)x`
• 
  `2ab = (2/3)xy`

We can work out the value of b:

• 
  `2ab = (2/3)xy = 2(1/2x)(2/3y)`
• 
  `b=(2/3)y`

Then we are missing b², what needs to be added:

• 
  `b^2=(2/3y)^2=4/9y^2`