Transcript Completing The Square - Rahway Public Schools

Completing The Square
Section 3.3 Beginning on page 112
The Big Idea
Completing the square is a technique we can use to write a quadratic in vertex form.
Completing the square is creating a Perfect Square Trinomial.
We would want the equation in vertex form so that we can…
• Solve the equation using square roots.
• Identify the vertex (possibly to identify the maximum or minimum).
The Perfect Square Trinomial
𝑎2 ± 2𝑎𝑏 + 𝑏2 = (𝑎 ± 𝑏)2
16𝑥 2 + 24𝑥 + 9
= (4𝑥 + 3)2
9𝑥 2 − 24𝑥 + 16
= (3𝑥 − 4)2
81𝑥 2 − 36𝑥 + 4
= (9𝑥 − 2)2
36𝑥 2 + 60𝑥 + 25 = (6𝑥 + 5)2
Solving Quadratics With Square Roots
Example 1: Solve 𝑥 2 − 16𝑥 + 64 = 100
𝑥−8
2
= 100
𝑥 − 8 = ±10
+8
+8
𝑥 = 10 + 8
𝑥 = −10 + 8
𝑥 = 18
𝑥 = −2
This is a perfect square
trinomial so we can factor it.
Completing The Square
If there is no perfect square trinomial, we can create something like on.
To complete the square we need to add
We must also subtract
𝑏 2
2
𝑏 2
2
after the b term.
to keep the equation balanced.
Example 3: Solving 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 when 𝑎 = 1
𝑏
2
𝑥 2 − 10𝑥 + 7 = 0
2
−10
=
2
2
= −5
𝑥 2 − 10𝑥 + 25 − 25 + 7 = 0
𝑥 = 5 ± 18
(𝑥 − 5)2 −18 = 0
5)2 =
(𝑥 −
18
𝑥 − 5 = 18
+5
+5
𝑥 =5± 9 2
𝑥 =5±3 2
2
= 25
Completing The Square
Example 4: Solving 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 when 𝑎 ≠ 1
3𝑥 2 + 12𝑥 + 15 = 0
3(𝑥 2 + 4𝑥 + 5) = 0
2
𝑥 + 4𝑥 + 5 = 0
When dividing both sides with that common factor,
you end up only with the reduced quadratic.
𝑏
2
2
4
=
2
2
= 2
2
=4
𝑥 2 + 4𝑥 + 4 − 4 + 5 = 0
(𝑥 +
2)2 +1
=0
2
(𝑥 + 2) = −1
𝑥 + 2 = −1
−2
−2
𝑥 = −2 ± −1
𝑥 = −2 ± 𝑖
Writing Quadratic Functions in Vertex
Form
Example 5: Write 𝑦 = 𝑥 2 − 12𝑥 + 18 in vertex form. Then identify the vertex.
𝑏
2
2
𝑦 = 𝑥 − 12𝑥 + 18
2
−12
=
2
2
= −6
𝑦 = 𝑥 2 − 12𝑥 + 36 − 36 + 18
𝑦 = (𝑥 − 6)2 −18 = 0
The vertex is at the point (6,-18)
2
= 36
Practice
Solve the equation using square roots:
1) 𝑥 2 + 4𝑥 + 4 = 36
2) 𝑥 2 − 6𝑥 + 9 = 1
3) 𝑥 2 − 22𝑥 + 121 = 81
Solve the equation by competing the square:
7) 𝑥 2 − 4𝑥 + 8 = 0
8) 𝑥 2 + 8𝑥 − 5 = 0
9) −3𝑥 2 − 18𝑥 − 6 = 0
10) 4𝑥 2 + 32𝑥 = −68
11) 6𝑥 𝑥 + 2 = −42
12) 2𝑥 𝑥 − 2 = 200
Write the quadratic function in vertex form. Then identify the vertex:
13) 𝑦 = 𝑥 2 − 8𝑥 + 18
14) 𝑦 = 𝑥 2 + 6𝑥 + 4
15) 𝑦 = 𝑥 2 − 2𝑥 − 6
Answers:
1) 𝑥 = 4 𝑎𝑛𝑑 𝑥 = −8
7) 𝑥 = 2 ± 2𝑖
10) 𝑥 = −4 ± 𝑖
13) 𝑦 = (𝑥 − 4)2 +2; (4,2)
2) 𝑥 = 4 𝑎𝑛𝑑 𝑥 = 2
8) 𝑥 = −4 ± 21
11) 𝑥 = −1 ± 𝑖 6
14) 𝑦 = (𝑥 + 3)2 −5; (−3, −5)
3) 𝑥 = 20 𝑎𝑛𝑑 𝑥 = 2
9) 𝑥 = −3 ± 7
12) 𝑥 = 1 ± 101
15) 𝑦 = (𝑥 − 1)2 −7; (1, −7)