Math III Unit 2: QUADRATIC MODELING AND EQUATIONS Main topics of instruction: 1) The Real Number System 2) Factoring and solving quadratic equations 3) Graphing quadratic equations 4) Complex Numbers Day 1: The Real Number System and Factoring There are two types of real numbers: _________________ numbers and _______________ numbers. Every real number can be graphed as a point on the number line. Rational Numbers Irrational Numbers
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Math III Unit 2: QUADRATIC MODELING AND EQUATIONS
Main topics of instruction:1) The Real Number System
2) Factoring and solving quadratic equations3) Graphing quadratic equations
4) Complex Numbers
Day 1: The Real Number System and Factoring
There are two types of real numbers: _________________ numbers and _______________ numbers. Every real number can be graphed as a point on the number line.
Rational Numbers Irrational Numbers
−4 −32 √5
Classify the following as rational or irrational. If a number is rational, state if it is a natural number, whole number, integer, or simply rational.
a) 4 b) -3 c) √6 d) 0.125
e) −25 f) √ 1
4g) 0 f) π
Critical Thinking: In each scenario, answer Always, Sometimes, or Never. If the answer is Sometimes, give examples of each outcome.
a) The sum of a rational number and a rational number is a rational number.
b) The product of two rational numbers is a rational number.
c) The sum of a rational number and an irrational number is an irrational number.
d) The product of a rational number and an irrational number is an irrational number.
e) The sum of two irrational numbers is an irrational number.
f) The product of two irrational numbers is an irrational number.
Factoring – Quadratics
Greatest Common Factor (GCF): ________________________________________________
These are called zeros! They are also called ________________________________________.
Factoring – Polynomials
Example 1: Factoring Using the GCF You try!
Factor and solve 2 x3−22x2+48 x=0 Factor and solve 3 x3+15 x2−42x=0
Example 2: Factoring Using Grouping You try!
Factor and solve 2 x3−3 x2−8x+12=0 Factor and solve 3 x2+12 x2−2 x−8=0
Day 2: Simplifying Radicals
means the _______________ of a number.
Consider √25. This means the square root of 25. To find it, ask yourself, "What number times itself equals 25?"
Evaluate.
1. 2. 3. 4. 5.
---------------------------------------------------------------------------------------------------------------------A radical is any quantity with a radical symbol, .
radical symbol'4' is the coefficient.Technically, 4 is beingmultiplied by 10.
'10' is the radicand.The radicand is thenumber "in the house".
Method #1 for Simplifying the Radicand - Perfect Squares
First, let’s make a list of important perfect squares!
Once again, one of the goals in simplifying radicals is to make the radicand as small as possible.
Example 1: Consider √12. What is the largest perfect square that multiplies into 12? ________
So, we can break √12 apart:
If a person had written , then no simplifying could be done, because 6 and 2 are not perfect squares.
You try! a) Simplify . Ask yourself, "Which of the perfect squares above divides evenly into 45?"
b) Simplify . c) Simplify .
Method #2 for Simplifying the Radicand - Twins and a Factor Tree
Example 2: Create a factor tree for 50: 50
Apply the story about the "twins" and the factor tree above in order to simplify .
If you want to use this method, you should always remember:1)As soon as a number kills its twin, it goes outside of the house IMMEDIATELY.2)If a number has no twins to kill, it must stay inside the house.3)All of the numbers inside and outside of the house are multiplied together in the end.
Simplify.
1. 2. 3.
When there are variables in the radicand, it is assumed that they represent positive values. In this situation, the "twins and a factor tree" method is very handy.
Nevertheless, consider √ x6. Since we are taking a square root, let’s break x6 up into as many x2 as possible.
Simplify.
4. √ x15 5. √48 y3 z4 6. 5√20x22
You try!
7. 8. 9.
Rationalizing a Denominator
Sometimes, we get radicals in the denominator of a fraction, but ______________
Example 1: Solve x2+6 x−7=0 by completing the square.
Step 1: Move the constant to the other side.
Step 2: Compute ( b2 )2
and add the result to both sides of the equation.
Step 3: Convert the left side to a binomial squared and simplify the right side.
Step 4: Square root both sides, and don’t forget the ± on the right side!
Step 5: Solve for x. Remember that the ± gives you two solutions!
Example 2:
1) Solve 2a2+12a+10=0 by completing the square.
You try! Solve the following by completing the square. (It’s okay to get decimals!)
a) n2+13n+22=7 b) 4 v2+16v=65
The Quadratic Formula
How many solutions? _____ How many solutions? _____ How many solutions? _____Type: __________________ Type: _________________ Type: __________________Discriminant is: __________ Discriminant is: __________ Discriminant is: __________
The quadratic formula is _________________________________________________________
What is the quadratic formula? Circle the discriminant!
Example 1: Use the discriminant to find the number and types of solutions to the quadratic expression. Remember to get all terms on one side and in standard form!
a) 3x2 – 5x - 18 b) 4x2 + 5 = 2x c) 2x2 = 3x – 12
Example 2: Use the quadratic formula to solve 3 x2−5 x=2. Then, state how many times and where the parabola would cross the x-axis.
You try! Use the quadratic formula to solve 5 x2+8 x−11=0. Then, state how many times and where the parabola would cross the x-axis.
You try! Use the quadratic formula to solve 9 x2−11=6 x. Then, state how many times and where the parabola would cross the x-axis.
Day 4: Complex Number Operations
You already know about real numbers (rational and irrational), but there are also ______________ numbers that use the letter ____.
i=¿i2=¿i3=¿i4=¿
Simplifying Using i
Example 1: Simplify √−8.
You try! Simplify the following: a) √−12 b) √−13Simplifying Complex Numbers
Standard Form of a Complex Number:
Example 2: Write √−9+6 in standard form.
You try! Write −√−50−2 in standard form.
Adding and Subtracting
Example 3: Simplify (5+7 i )+(−2+6 i)
You try! a) Simplify (−4+6 i )+(3−2 i) b) Simplify (8+3 i )−(2+4 i)
Multiplying Complex Numbers
Example 4: Simplify (5 i)(−4 i).
You try! Simplify (12 i)(7 i ).
Example 5: Simplify (2+3 i)(−3+5 i). F:O:I:L:
You try! a) Simplify (6−5 i)(4−3i ). b) Simplify 3 i(9−4 i). c) Simplify (8−2i )2.
Rationalizing
There is one big rule for complex number, and that is that _______________________________
How can I find the y value of the vertex? ____________________________________Example 1: Find the vertex and axis of symmetry,then graph y=x2+2 x+3
You try! Find the vertex and axis of symmetry, then graph y=−2x2+6 x−4 .
Finding a Quadratic Equation in Standard Form
Example 2: A parabola has three points: (2, 3), (3, 13), and (4, 29). Find a quadratic equation (model) in standard form that will fit the parabola.
You try! A parabola has three points: (1, 0), (2, -3), and (3, -10). Find a quadratic equation (model) in standard form that will fit the parabola.
Example 3: Anthony throws a football across the field while standing on top of the bleachers. The data that follows gives the height of the ball in feet versus the seconds since the ball was thrown.
Write a quadratic model for this data. (Round to two decimal places.)