Question

Complete the square: ax^2 + x + 3.

Answer 1

`a(x+(1)/(2a))^2)-(1)/(4a)+3`

Explanation:

We have been given an expression
`ax^2+x+3`. We are asked to complete the square for the given expression.

First of all, we will factor our a as:

`a(x^2+(x)/(a))+3`

`a(x^2+(1)/(a)x)+3`

Now, we need to add and subtract half the square of the middle term, that is
`((1)/(2a))^2`:

`a[x^2+(1)/(a)x+((1)/(2a))^2-((1)/(2a))^2]+3`

`a[x^2+(1)/(a)x+((1)/(2a))^2-((1)/(4a^2))]+3`

`a(x^2+(1)/(a)x+((1)/(2a))^2)+3-a*(1)/(4a^2)`

`a(x+(1)/(2a))^2)-(1)/(4a)+3`

Therefore, our required square would be
`a(x+(1)/(2a))^2)-(1)/(4a)+3`.

Answer 2

The value of a is
`(1)/(12)`.

Explanation:

The given expression is

`ax^2+x+3`

A quadratic expression
`ax^2+bx+c` is complete square if
`b^2-4ac=0`

For the given expression a=a,b=1 and c=3.

`(1)^2-4(a)(3)=0`

`1-12a=0`

Add 12a on both sides.

`1=12a`

Divide both sides by 12.

`(1)/(12)=a`

Therefore, the value of a is
`(1)/(12)`.

`(1)/(12)x^2+x+3`

`((1)/(2√(3))x)^2+2((1)/(2√(3))x)(√(3)+(√(3))^2`

`((1)/(2√(3))x+√(3))^2`
`[\because (a+b)^2=a^2+2ab+b^2]`