Lesson 2 – Completing the Square
Recall
Standard Form
f(x) = ax2 + bx + c
Vertex Form
f(x) = a(x – h)2 +k
Example 1 Convert each of the following into standard form.
a) f(x) = –3(x + 1)2 – 5
b) f(x) = ½ (x – 6)2 + 3
Example 2 Factor each of the following:
a) x2 + 12x + 36
Completing the Square:
b) x2 – 14x + 49
Adding a constant to a quadratic expression to form a
perfect square trinomial.
Example 3 Identify the values of a, b, and c, in each of the following:
a) 3x2 – 8x + 4
b) -3 +4x – x2
Example 4 The Process of Completing the Square:
a=1
Complete the Square:
x2 – 8x + 4
1
Write the value of 𝑏 = ________
Divide b by 2 and square the result
𝑏 2
(2) = ______
a≠1
Complete the Square:
2
Add and subtract the value from 1
3
Factor the first three terms (it will be a
perfect square trinomial)
1
Factor 𝑎 from the first and second terms
ONLY.
2
Write the NEW value of 𝑏 = ________
Divide b by 2 and square the result
2x2 – 8x + 3
𝑏 2
(2) = ______
3
Add and subtract the value from 1
4
Multiply the last term by ‘a’
5
Factor the three terms inside the brackets
it will be a perfect square trinomial)
Example 5 Determine the maximum/minimum value of each of the following:
1
a) 𝑓(𝑥) = – 𝑥 2 + 𝑥 – 5
b) 𝑓(𝑥) = – 4𝑥 2 − 4𝑥 – 5
2