Question

What value completes the square for the expression? x² – 12x

Answer 1

Some complete square will look like
\[(x+a)^2\\~\\\text{if we expand this}\\~\\=(x+a)(x+a)\\=x(x+a)+a(x+a)\\=x^2+ax+ax+a^2\\~\\\text{we get}\\~\\=x^2+2ax+a^2\]

Answer 2

so how do i start this? im still really confused

Answer 3

or, for \(b=2a\), ie \(a=\frac b2\)

\[(x+\tfrac b2)^2=x^2+bx+(\tfrac{b}{2})^2\]

Answer 4

you kinda have to use this idea, backwards

Answer 5

You have
\[x^2-12x\]
Compare this with
\[x^2+bx+(\tfrac b2)^2\]

Answer 6

The value that completes the square is equal to \((\frac b2)^2\)

Answer 7

So identify \(b\) from your equation,

Answer 8

-12

Answer 9

right, so
now
compute \((\tfrac b2)^2\)

Answer 10

36

Answer 11

good, we now have
\[x^2-12x+36\]
let's check that it completes the square

Answer 12

if we break up that middle term in half
\[=x^2-6x-6x+36\]
and reverse the expansion
\[=x(x-6)-6(x-6)\\
=(x-6)(x-6)\\
=(x-6)^2\]

Answer 13

Does this start to make sense?

Answer 14

yes!

Answer 15

if you had an equation to start with, say
\[x^2-12x= 100\]
Completing the square would show
\[(x-6)^2=136\]