Algebra: Quadratics - Completing the Square

Worked Solution:

( c + a )² = c² + 2 a c + a²
( c + a )² + b = c² + 2 a c + a² + b
We want to find values for a and b to make the line above into c² – 12 c – 1
We already have the c² term so let's look at the c term:

2a c =

–12

c :

For the quadratic in the question, the c coefficient is

–12

which must be equal to the c term above:

2 a =

–12

a =

–6

a2 + b =

–1

: The quadratic given has a constant term of

–1

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

–6

a2 =

36

:

36

+ b =

–1

b = –37

y =

c2 – 12 c – 1

= (

c – 6

)2 –37

The squared term is always positive or zero

the least value of y must be –37 when the squared term is zero: (

c – 6

)2 = 0

c =

6

The turning point is therefore (6, –37)