Algebra: Quadratics - Completing the Square

Worked Solution:

( p + a )² = p² + 2 a p + a²
( p + a )² + b = p² + 2 a p + a² + b
We want to find values for a and b to make the line above into p² + 12 p – 2
We already have the p² term so let's look at the p term:

2a p =

12

p :

For the quadratic in the question, the p coefficient is

12

which must be equal to the p term above:

2 a =

12

a =

6

a2 + b =

–2

: The quadratic given has a constant term of

–2

This must equal a2 + b and we now know the value of a =

6

a2 =

36

:

36

+ b =

–2

b = –38

y =

p2 + 12 p – 2

= (

p + 6

)2 –38

The squared term is always positive or zero

the least value of y must be –38 when the squared term is zero: (

p + 6

)2 = 0

p =

–6

The turning point is therefore ( –6, –38)