Question

y = x2 - 20x + 2 can be written in the form y = (x + a)^2 + b.
Find what is a and b.

Answer 1

Completing the square is a useful technique for solving quadratic equations. It is a more powerful technique than factorisation because it can be applied to equations that do not factorise.
When completing the square, an expression like,
ax2 + bx + c is written in the form (Ax + B)2 + C.
We will begin with the simple example where a = 1. In this case we will write expressions in the form
x2 + bx + c as (x + B)2 + C
If we expand (x + B)2 + C we get x2 + 2Bx + B2 + C.
Comparing this with x2 + bx + c shows that
b = 2B and c = B2 + C
which gives B =
b
2
and C = c – B2
Using these two results we can now set about completing the square in some simple cases.