Q UA D R AT IC S
ANGRY BIRDS!
Angry Birds follow a parabolic path. Your quest is to find the equation of the
parabola to cause the angry bird to hit the pig. Use the given information and
the quadratic regression method to find the equation of the curve.
All numbers are decimals – round to the thousandths place.
When you think your answer is correct, bring it to me to check it on the computer.
Groups that successfully find all four equations will earn a prize. (woooo!)
Homework:
page 245 (1-19, 33-37) odd
5-2 PROPERTIES OF PARABOLAS
MAXIMUM OR MINIMUM
homework:
5-4 FACTORING QUADRATIC EXPRESSIONS
EQ: HOW DO YOU REDUCE A QUADRATIC
EXPRESSION INTO ITS LINEAR FACTORS?
Factor these expressions – Algebra 1 Review – Do you
remember?
5-4 FACTORING QUADRATIC EXPRESSIONS
EQ: HOW DO YOU REDUCE A QUADRATIC
EXPRESSION INTO ITS LINEAR FACTORS?
• Factoring is rewriting an expression as the
product of its factors.
• The greatest common factor (GCF) of the
expression is a common factor of the term of
the expression.
5-4 FACTORING QUADRATIC EXPRESSIONS
EQ: HOW DO YOU REDUCE A QUADRATIC
EXPRESSION INTO ITS LINEAR FACTORS?
When you factor a quadratic expression in the form ax2 + bx +c you are looking for a
pair of factors that multiply to equal ac and add to equal b.
5-4 FACTORING QUADRATIC EXPRESSIONS
EQ: HOW DO YOU REDUCE A QUADRATIC
EXPRESSION INTO ITS LINEAR FACTORS?
5-4 FACTORING QUADRATIC EXPRESSIONS
EQ: HOW DO YOU REDUCE A QUADRATIC
EXPRESSION INTO ITS LINEAR FACTORS?
5-4 FACTORING QUADRATIC EXPRESSIONS
EQ: HOW DO YOU REDUCE A QUADRATIC
EXPRESSION INTO ITS LINEAR FACTORS?
5-4 FACTORING QUADRATIC EXPRESSIONS
EQ: HOW DO YOU REDUCE A QUADRATIC
EXPRESSION INTO ITS LINEAR FACTORS?
Homework: page 268 (1-45) every other odd
5-4 SOLVING QUADRATIC EQUATIONS
Solving a quadratic equation means finding the values of the variable that make
the equation true.
Usually, for a quadratic equation, there are two solutions.
There are several methods to solve quadratic equations:
• Factoring
• Finding Square Roots
• Completing the Square
• Using the Quadratic Formula
5-4 SOLVING QUADRATIC EQUATIONS
Solving by factoring requires:
Setting the equation equal to zero
Completely factoring the equation
Using the Zero-Product property to find the zeros.
ZERO PRODUCT PROPERTY: If ab = 0 then a = 0 or b = 0
Example: if 𝒙 + 𝟑 𝒙 − 𝟕 = 𝟎 then 𝒙 + 𝟑 = 𝟎 𝒐𝒓 𝒙 − 𝟕 = 𝟎
Set each factor equal to zero and solve for the variable. This solution is called a
zero of the equation because it makes the equation equal zero.
5-4 SOLVING QUADRATIC EQUATIONS
ZERO PRODUCT PROPERTY: If ab = 0 then a = 0 or b = 0
Example: if 𝒙 + 𝟑 𝒙 − 𝟕 = 𝟎 then 𝒙 + 𝟑 = 𝟎 𝒐𝒓 𝒙 − 𝟕 = 𝟎
Standard Form for a Quadratic Equation: 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎
𝒙𝟐 − 𝟓𝒙 = 𝟐𝟒
𝟐𝒙𝟐 − 𝟏𝟏𝒙 + 𝟓 = −𝟏𝟎
5-4 SOLVING QUADRATIC EQUATIONS
Use factoring to solve the following equations:
1) 𝒙𝟐 + 𝟕𝒙 = 𝟏𝟖
2) 𝟐𝒙𝟐 + 𝟒𝒙 = 𝟔
3) 𝟏𝟔𝒙𝟐 = 𝟖𝒙
5-4 SOLVING QUADRATIC EQUATIONS
Solving by finding Square Roots is used when there is no linear term.
• Rewrite equation as 𝒂𝒙𝟐 = 𝒄
• Isolate 𝒙𝟐
• Find square roots (remember, there are two!)
• Example: 𝟓𝒙𝟐 − 𝟏𝟖𝟎 = 𝟎
Try These:
𝟒𝒙𝟐 − 𝟐𝟓 = 𝟎
𝟑𝒙𝟐 = 𝟐𝟒
𝟏
𝟒
𝒙𝟐 − = 𝟎
SIMPLIFYING SQUARE ROOTS
𝑎𝑏 =
𝑎 ∙
𝑏
𝑎
𝑎
=
𝑏
𝑏
𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 48
48 = 16 ∙
=4 3
3
SIMPLIFYING SQUARE ROOTS
Break the number in the radical down to its prime factors
– use a factor tree or repeated division.
72 = 9 ∙ 8 = 3 ∙ 3 ∙ 4 ∙ 2 = 3 ∙ 3 ∙ 2 ∙ 2 ∙ 2
Each pair of factors represents a single root that you can
solve out of the radical
𝟕𝟐 = 𝟑 ∗ 𝟑 ∗ 𝟐 ∗ 𝟐 ∗ 𝟐 = 𝟑 ∗ 𝟐 𝟐= 6 𝟐
SIMPLIFYING SQUARE ROOTS
Process is true for variables as well
Every pair of variables represents a single root variable
𝒃𝟑 =𝒃 𝒃
5-6 COMPLEX NUMBERS
HOW DO YOU TAKE THE SQUARE ROOT OF A NEGATIVE
NUMBER?
Up until now, there was no way to deal with a root like this: −𝟐𝟓
The letter i is defined as the square root of negative 1, and can be simplified out of a
square root.
The numeral is rationalized in the usual way
−𝟐𝟓 =
−𝟑𝟔
−𝟏𝟎𝟎
−𝟓𝟎
−𝟒𝟖
−𝟏 (𝟐𝟓) = 𝒊 𝟐𝟓 = 𝟓𝒊
5-6 COMPLEX NUMBERS
HOW DO YOU TAKE THE SQUARE ROOT OF A NEGATIVE
NUMBER?
5-6 COMPLEX NUMBERS
HOW DO YOU TAKE THE SQUARE ROOT OF A NEGATIVE
NUMBER?
Use the Complex Number Plane to represent a complex number
geometrically.
Locate the real part of the complex number on the horizontal axis
and the complex part on the vertical axis.
𝟏 + 𝟑𝒊
−𝟐 − 𝟐𝒊
−𝟑𝒊
5-6 COMPLEX NUMBERS
HOW DO YOU TAKE THE SQUARE ROOT OF A NEGATIVE
NUMBER?
The absolute value of a complex number is its distance from the origin in the complex
number plane.
You can find the absolute value by using the Pythagorean Theorem.
𝒂 + 𝒃𝒊 =
Find the absolute value:
𝟑 + 𝟒𝒊
−𝟐 − 𝟓𝒊
𝒂𝟐 + 𝒃𝟐
5-6 COMPLEX NUMBERS
HOW DO YOU TAKE THE SQUARE ROOT OF A NEGATIVE
NUMBER?
When you add or subtract complex numbers you combine the real parts and
imaginary parts separately.
When you multiply complex numbers you use the rules for multiplying binomials (FOIL)
Remember that i2 = -1
5-6 COMPLEX NUMBERS
HOW DO YOU TAKE THE SQUARE ROOT OF A NEGATIVE
NUMBER?
5-6 COMPLEX NUMBERS
HOW DO YOU TAKE THE SQUARE ROOT OF A NEGATIVE
NUMBER?
5-6 COMPLEX NUMBERS
HOW DO YOU TAKE THE SQUARE ROOT OF A NEGATIVE
NUMBER?
Write each answer in a + bi form
5-7 COMPLETING THE SQUARE
USING PERFECT SQUARES TO SOLVE
EQUATIONS
5-7 COMPLETING THE SQUARE
USING PERFECT SQUARES TO SOLVE
EQUATIONS
You can solve an equation where one side of the equation is a perfect square by
finding square roots.
5-7 Completing the Square
Using perfect squares to solve equations
If one side of an equation is not a perfect square, you can rewrite the constant
term to get a perfect square trinomial.
Use this relationship to complete the square:
𝟐
𝒃
𝒃
𝒙𝟐 + 𝒃𝒙 +
= (𝒙 + )𝟐
𝟐
𝟐
𝒙𝟐 + 𝟏𝟎𝒙 +
5-7 Completing the Square
Using perfect squares to solve equations
Find the missing constant to complete the square: write the factored square.
5-7 Completing the Square
Using perfect squares to solve equations
5-7 Completing the Square
Using perfect squares to solve equations
5-7 Completing the Square
Using perfect squares to solve equations
5-7 Completing the Square
Using perfect squares to solve equations
5-7 Completing the Square
Using perfect squares to solve equations
5-7 Completing the Square
Using perfect squares to solve equations
homework: page 289 (1-33) odd
Chapter 5 study guide will be given out at our next class.
Chapter 5 Test will be given the Thursday(5th) and Friday (6th) after Thanksgiving
break.
5-8 THE QUADRATIC FORMULA
Homework:
p 289 (23-33) odd
p 297 (1-39) odd