Completing
the Square
Factoring review The factoring that you have learnt thus far allows you to find the zeros of a quadra9c func9on (AKA second degree polynomial) Completing the square is
another method used to solve
quadratic equations.
It allows you to identify the
vertex of a parabola when
graphing.
The general form of a
quadratic equation is:
2
ax
+ bx + c = 0,
a≠0
The factored form of a
quadratic equation is:
a(x + x1)(x + x2)= 0,
a≠0
Where x1 and x2 are the zeros
of the function
The standard form of a
quadratic equation is:
a(x –
2
h)
+ k = 0,
a≠0
Where (h,k) is the vertex of
the function
Step 1
Write the equation
in the form
2
ax
+ bx ___ +c = 0
Step 2
If a ≠ 1, divide the first
two term of the equation
by a.
Step 3
Complete the Square
– Take ½ of b, and
square the result.
– Add and subtract this result to the
equation
Step 4
2
(b/a)
Remove
from the
brackets (remember to multiply
it by a)
Step 5
Factor the trinomial left in
the brackets as a perfect
square
Step 6
Simplify the term(s)
left outside of the
brackets
Comple9ng the square f(x) = ax2 + bx + c
f(x) = x2 -6x + 8
1. factor out a from the first
two terms ONLY
2. complete the square of
the first two terms
3. write your answer in
standard form
Comple9ng the square f(x) = ax2 + bx + c
f(x) = x2 -6x + 8
f(x) = x2 -6x + 8
f(x) = (x2 -6x
)+8
1. factor out a
2. complete the square
3. re-write in standard form
Comple9ng the square f(x) = ax2 + bx + c
f(x) = x2 -6x + 8
f(x) = x2 -6x + 8
f(x) = (x2 -6x
)+8
f(x) = (x2 -6x +9 - 9) + 8
f(x) = (x2 -6x +9) -9 + 8
f(x) = (x2 -6x +9) - 1
1. factor out a
2. complete the square
3. re-write in standard form
Comple9ng the square f(x) = ax2 + bx + c
f(x) = x2 -6x + 8
f(x) = x2 -6x + 8
f(x) = (x2 -6x
)+8
f(x) = (x2 -6x +9 - 9) + 8
f(x) = (x2 -6x +9) -9 + 8
f(x) = (x2 -6x +9) - 1
f(x) = (x -3)2 - 1
1. factor out a
2. complete the square
3. re-write in standard form
Complete the Square Examples 1.  x2 + 5x + 6 8.  x2 + 13x + 12 2.  x2 − 6x + 4 9.  x2 + 8x + 12 3.  x2 + 10x – 24 10. x2 − 8x – 9 4.  x2 + 3x – 1 11. x2 + 13x – 48 5.  x2 + 6x + 8 12. x2 − 20x + 64 6.  x2 + 4x – 5 13. x2 − 7x – 4 7.  x2 − 7x + 10 14. x2 − 2x − 3 Answers
Comple9ng the square – Difficult f(x) = ax2 + bx + c
f(x) = 4x2 + 8x -60
1. factor out a from the first
two terms ONLY
2. complete the square of
the first two terms
3. write your answer in
standard form
Comple9ng the square f(x) = ax2 + bx + c
f(x) = 4x2 + 8x -60
f(x) = 4x2 + 8x -60
f(x) = 4(x2 + 2x) - 60
1. factor out a
2. complete the square
3. re-write in standard form
Comple9ng the square f(x) = ax2 + bx + c
f(x) = 4x2 + 8x -60
2
f(x) = 4x + 8x -60
2
f(x) = 4(x + 2x) -60
2
f(x) = 4(x2 + 2x +2 -2) -60
f(x) = 4(x + 2x +2) -8 -60
1. factor out a
2. complete the square
3. re-write in standard form
Comple9ng the square f(x) = ax2 + bx + c
f(x) = 4x2 + 8x -60
2
f(x) = 4x + 8x -60
2
f(x) = 4(x + 2x) -60
2
f(x) = 4(x 2 + 2x +2 -2) -60
f(x) = 4(x + 2x +2) -8 -60
f(x) = 4(x + 1)
2
-68
1. factor out a
2. complete the square
3. re-write in standard form