Question

What should be added to the polynomial so that it becomes perfect square 4X square plus 8X?

`4x^2+8x`

Answer 1

4

Explanation:

By adding four we can make it perfect square
`4x^2 + 8x + 4`
`(2x)^2 + 2*2x*2 + (2)^2`
`(2x + 2)^2`
Thus it becomes a perfect square

But how do we get to this answer without any guess , here's the role of discriminant.
Any quadratic equation is of form
`ax^2+bx+c=0`
Here , discriminant `(d)=b^2-4ac`
For perfect squares it is `0`
So , we get
`0=8^2-4×4×c`
`64=16c`
`c=4`
That's the answer

Answer 2

`4x^2+8x+4` so you add 4

Explanation:

A perfect square is of form: `(a+b)^2=a^2+2ab+b^2`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Let the unknown value be `t` ( you have to write something for it!). `t` could be just a number and or a coefficient of `x->?x`

Set `4x^2+8x+t color(white)("ddd")=color(white)("ddd")a^2+2ab+b^2`

Compare `a^2=4x^2->2^2x^2color(white)("d")"so "color(blue)(a =sqrt(2^2x^2)=2x)`

Compare `2ab=8x`

Substituting for `color(red)(a)` we have:

Compare `color(green)(2color(red)(a)b=8xcolor(white)("d")->color(white)("d")2(color(red)(2x))b=8x color(white)("d")->color(white)("d")4xb=8x)`

`color(blue)("so "b=2)`

Thus `a^2+2ab+b^2color(white)("d")->color(white)("d")4x^2 +(2)(2x)(2)+2^2`

Giving:

`color(white)("ddddddddd") color(blue)(ul(bar(|color(white)(2/2) 4x^2+8x+4 color(white)(2)|)))`