Quadratic Formula & Completeing the Square
Name:
_____________________
Simplifying Radicals
Perfect Square numbers are 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225…
You have to factor out the largest perfect square that the radical is divisible by
Examples:
1) √75
2) √40
3) √242
4)√72
2) 500
3) √50
4) √18
6) 84
7)√125
8) 288
√𝟐𝟓√𝟑
𝟓√𝟑
Try these:
1)
24
5) 28
Solving radical equations:
1) 2x2 = 50
2) x2 – 8 = 40
3) 2x2 = 54
2) 6x2 = 120
3) -7x2 = -525
Try These:
1) x2 + 9 = 58
23
Simplifying Complex Radicals:
1) What number times itself equals 9? ___
2) What number times itself equals -9? ___
We call square roots of negative numbers complex radicals.
Since we know the √9 = 3, then we can use “imaginary numbers” to find that √−9 equals 3i
Examples:
1) √−25
2) √−121
3) √−44
4) √−45
9) √−128
10) √−242
11) √−81
Try these:
8) √−32
Solving radical equations:
𝑥2
= -16
1) -2p2= 250
2)
4) -7m2 = 49
5) x2 – 9 = -37
3
3) y2 + 10 = -26
6) r2 = -121
24
Practice with simplifying:
Simplifying with perfect squares:
6 ±√144
1)
2
2)
−3 ±√289
5
3)
5 ±√81
2
4)
10 ±√625
5
Simplify the following square roots:
5) √20
6) √125
7) √392
8) √48
Simplifying with square roots that can be simplified:
9)
2 ±√20
13)
6
3 ±√18
6
10)
14)
20 ±√125
10
30 ±√200
10
11)
15)
21 ±√392
14
21 ±√45
24
12)
16)
12 ±√48
4
20 ±√75
5
25
The Quadratic Formula
A quadratic function is standard form is written __________________.
The highest power in a quadratic function is _____.
The quadratic formula:
𝒙=
−𝒃 ∓ √𝒃𝟐 −𝟒𝒂𝒄
𝟐𝒂
** Can be used to factor quadratic functions **
Rules:
1) Make sure the equation is set equal to zero
2) Identify “a”, “b”, and “c”
3) Plug numbers into the quadratic formula
4) Simplify
Examples:
1) x2 + 3x – 4 = 0
2) 9x2 – 6x – 11 = 0
a = _____ b = ______ c = _____
a = _____ b = ______ c = _____
x=
−𝑏 ± √𝑏 2 −4𝑎𝑐
2𝑎
=
x=
3) 2a2 − a − 15 = 0
4) 4x2 + 11x – 20 = 0
a = _____ b = ______ c = _____
a = _____ b = ______ c = _____
x=
x=
5) x2 – 3x – 3 = 0
6) x2 + 5x + 5 = 0
a = _____ b = ______ c = _____
a = _____ b = ______ c = _____
x=
x=
26
Solve for
ax2
+ bx + c = 0, then factor using the quadratic formula:
𝒙=
−𝒃 ∓ √𝒃𝟐 − 𝟒𝒂𝒄
𝟐𝒂
1) 4x2 – 1 = -8x
2) 4x2 + 7x = 15
a = _____ b = ______ c = _____
a = _____ b = ______ c = _____
x=
x=
3) 2x2 + 23 = 14x
4) 2x2 + 39 = -18x
a = _____ b = ______ c = _____
a = _____ b = ______ c = _____
x=
x=
5) 5x2 + 3x = -1
6) 5x2 + 125 = -50x
a = _____ b = ______ c = _____
a = _____ b = ______ c = _____
x=
x=
27
Quiz Review:
−𝒃 ∓ √𝒃𝟐 − 𝟒𝒂𝒄
𝟐𝒂
𝒙=
Simplify the following:
1) √162
2) √−150
3)
14±√121
2
4)
12±√128
20
Solve the following for x:
5) 8x2 = 96
6) x2 – 10 = -46
7)
x2
4
= 10
8) -12x2= 192
Solve using the quadratic formula:
9) 8x2 – 8x – 30 = 0
10) 9x2 + 4x = 16
_________________=____
a = _____ b = ______ c = _____
a = _____ b = ______ c = _____
x=
x=
28
Completing the Square
Examples:
Problem
Move the constant to the
right hand side**
Prepare to add the
needed value
x2 + 8x – 4 = 0
x2 + 6x + 1 = 0
x2 – 8x + 15 = 0
___ ÷ 2 = __ 𝑎𝑛𝑑 ___2 = ____
___ ÷ 2 = __ 𝑎𝑛𝑑 ___2 = ____
x2 + 8x = 4
x2 + 8x + ____= 4 + ____
Take half and square the
coefficient of x**
8 ÷ 2 = 4 𝑎𝑛𝑑 42 = 16
Add value to both sides
x2 + 8x + 16 = 4 + 16
x2 + 8x + 16 = 20
Factor left side
(x + 4)(x + 4)=20
(x + 4)2 = 20
Solve problem by taking
square root of both sides
√(𝑥 + 4)2 = √20
𝑥 + 4 = ±2√5
Steps marked ** may be skipped!
𝑥 = −4 ± 2√5
Try these:
1) x2 + 10x + 21 = 0
2) x2 – 12x + 23 = 0
3) x2 + 6x – 59 = 0
29
Solve the problems by completing the square:
1) x2 + 14x – 15 = 0
2) x2 – 4x + 6 = 0
3) x2 + 6x + 8 = 0
4) x2 +14x – 51 = 0
5) x2 + 2x + 20 = 0
6) x2 + 24x – 16 = 0
30