Quadratics
Part 1
Angela and Sarah
Belmont Free Lessons
Table of contents
Homework
Absolute Value Equations
Quadratic Standard Form
Basic Factoring
We’ll go over the homework from last week and answer questions!
We’ll talk about what absolute value is and what the equations look like when graphed
We’ll introduce quadratics and the basic structure
We’ll explain how to factor some of the quadratics you may see in Algebra 1
01
02
03
04
01
Homework
We’ll go over the solutions for the homework due today. Please ask us any questions if you have them in the chat
Homework Solutions
53 = 5 * 5 * 5 = 125
Use the distributive property to get: 12u2 - 12uv3
Binomial because there are two terms: 3x2y and 7y
Homework solutions continued
4. What degree is associated with the polynomial 2x3-6x+3?
5. What type of polynomial is 7x2y3z4?
6. What degree is associated with the polynomial 7x5-6x4-2x3+4x2-3x+7
7. What type of polynomial is 2x-4y+3z?
Degree 3 (x3); Cubic
Monomial (only one term!)
Degree 5 (x5); Quintic
Trinomial (three terms)
Homework solutions continued
8. Expand -2(x+2y-3z)
9. Expand 25a2(5a)
10. Is the point shown a minimum or a maximum?
Use the distributive property to get: -2x-4y+6z
We can break it up into 25*5*a2*a = 125a3
Maximum because it is the highest point of the curve
Before we get into the topics for today, let’s go over the absolute value material that we didn’t get to cover last week!
Absolute Value Equations
02
DOES ANYONE KNOW WHAT AN ABSOLUTE VALUE FUNCTION LOOKS LIKE ON A GRAPH?
What is absolute value?
Absolute value is the distance a number is from zero on a number line without considering the direction.
Remember that the absolute value of a number is always positive!
In the picture above, you can see that the distance between -3 and 0 is 3. The Absolute Value of -3 would then be 3 because the distance is positive 3 units.
Try some on your own!
Answers:
Solutions for 3 and 4
3. |x| + 2 = 4
|x| = 2, x = ±2
(x = 2, x = -2)
4. |x+7| = 12 We can break this problem into two parts
x+7 = 12 x+7 = -12
x = 5 x = -19
One tip for checking your work is to plug these x values back into the original equation to see if you get 12 in the end
More practice!
Answers:
Choose one of these for us to go over!
How do we graph absolute value functions?
Say you have the function:
f(x) = |x|
And you make a chart with 5 different points and their x and y values. You can plot these points On a graph to get something that looks like the picture on the right. Since absolute value of a number is always positive, the function is never negative on the graph.
Important points
(remember this!)
The axis of symmetry is line that
evenly divides the graph into two parts (the middle of the graph). In this case, because the function is f(x) = |x|, the aos is zero, or on the y-axis
(absolute value is always positive)
Other examples
What if you flipped the graph?
We kept saying that absolute value is always positive, but if the function has a negative in front (-|x| instead of |x|), the whole graph is going to be negative.
When you turn everything negative, all the values that used to be positive are now negative. That’s why the graph flips; to check this, find a point on the blue line and drag your fingers/eyes vertically to the red line to find that point as negative (ex: x = 1 becomes x = -1).
Other examples
What if you changed the coefficient in front of the absolute value?
It’s the same as slope! The greater the coefficient, the steeper the line will be.
Similarly, the smaller the coefficient, the wider the line will be. In the graph, y = 2|x| is the same as y = ½ |x| but just a lot steeper! In algebra, they call this stretching (getting steeper) and compression (getting wider)
You’ll go deeper into absolute value functions and how to graph them in future algebra 1 and 2 classes. We just wanted to give you an introduction to what they look like so you can recognize them in the future!
Quadratic Standard Form
03
What are Quadratics?
Quadratics is arguably the most important topic in Algebra 1 along with Functions. This is because knowing how to solve quadratic equations can be very useful in real life problems!
You might think that’s really strange since we associate quad with 4, but just remember that quadratic equations always have a term with x2
Quadratics
The name Quadratic comes from "quad" meaning square, because the variable gets squared (like b).
Quadratic Standard Form
a, b and c are known values. a can't be 0.
Examples:
b = −3
And where is c? Well c=0, so is not shown.
ax2+bx+c = 0
But sometimes it’s not that simple!
You’ll see problems that aren’t in that form already
Move all the terms to one side:
x2-3x+1 = 0 (now it’s in standard form)
Expand to get: 2w2-4w = 5
Move all the terms to one side:
2w2-4w-5 = 0
Expanding Quadratics: the FOIL method
Sometimes, you’ll be asked to expand a quadratic that is written as the product of two linear expressions.
What does that mean?
It’s usually something like this: (x-4)(x+3)=0
FOIL stands for:
First�Outside
Inside
Last
(x-4)(x+3)
x2
(x-4)(x+3)
3x
(x-4)(x+3)
-4x
(x-4)(x+3)
-12
Final Answer:
x2+3x-4x-12
=x2-x-12
Try expanding these quadratics!
(x+1)(x+2)=0
(x-1)(x+2)=0
Challenge: (2x+1)(x-3)=0
x^2 + 3x + 2
x^2 + x - 2
2x^2 - 5x - 3
04
Basic Factoring
Factoring
Numbers have factors:
2 * 3 = 6
factor factor
And expressions (like x2+4x+3) also have factors:
(x+3)(x+1) = x2+4x+3
factor factor
Factoring explained
Factoring: Finding what to multiply together to get an expression. It is like "splitting" an expression into a multiplication of simpler expressions.
Example: factor 2y + 6
Both 2y and 6 have a common factor of 2: (2y is 2 × y and 6 is 2 × 3)
So we can factor the whole expression into:
2y+6 = 2(y+3)
So 2y+6 has been "factored into" 2 and y+3
You can also imagine it like the opposite of expanding an expression
Common factors
In the previous example we saw that 2y and 6 had a common factor of 2
But for quadratics we need the highest common factor, including any variables
Example: factor 3y2+12y
We first notice that both 3y2 and 12y have a common factor of 3, but we need the highest common factor.
3y2+12y becomes 3(y2+4y) Can we factor this even further?
Yes! y2 and 4y both have a common factor of y. The highest common factor is 3y because both terms have 3y (we combine the 3 and the y)
The expression becomes: 3y(y+4)
Factoring basic quadratics
This is where it gets a bit tough, but once you understand how to approach these equations solving these problems will become more manageable.
Example: factor x2-2x-3
This might look impossible at first (none of the terms have common factors!), but we can break it down to help you understand.
Let’s assume that we can factor x2-2x-3 into this expression (x+a)(x+b). (a and bare numbers)
If we expand (x)(x), what do we get?
We get x2-bx-ax+ab (remember FOIL!)
Can you guys try factoring -bx-ax? What’s the common factor?
-x(b+a)
Cont.
x2-2x-3 = (x+a)(x+b)
a*b= -3, a+b= -2
Can you find two numbers that would fit both equations?
This is a bit like guessing, but the two numbers that would work are -3 and 1!
We can factor it as (x-3)(x+1)!
Let’s double check this by expanding (x-3)(x+1):
x2+x-3x-3 = x2-2x-3
Tada! It’s a bit like solving a puzzle.
Let’s try another one!
x2+4x+3
We know that x2+4x+3 = (x+a)(x+b)
a*b= 3, a+b= 4
Can you guys find two numbers that make this true?
The only factors of 3 and 1 and 3, so we can determine that a and b can only be either 1 or 3.
Our factored expression: (x+1)(x+3)
Your turn!
Try factoring:
x2+x-2 = (x+a)(x + b) a*b= -2, a+b= 1
factors: 2 and -1 Answer: (x-2)(x+1)
x2-4x-21 = (x + a)(x + b) a*b= -21, a+b= -4
factors: 3 and -7 Answers: (x-3)(x+7)
Example Problem
Now let’s look at one that’s a little more difficult:
z3 − z2 + 9z - 9
This might look like a lot at first, but we can split this expression into two parts
z3 − z2 and 9z - 9
We notice that
z3 and z2 have a common factor Type in the chat what you think it is
and that − 9z and 9 have a common factor Type in the chat what you think it is
Common Factors continued...
z3 − z2 + 9z - 9
Common factor of z3 and z2: z2 Common factor of 9z and -9: 9
So we end up with z2(z-1) + 9(z-1)
We notice that both terms have a z-1. Turns out that you can actually combine the two (z2 with 9 and the two (z-1)’s)
This is what we end up with: (z2+9)(z-1)
Remember that your final answer should always be completely factored; you should find a way to continue factoring or your answer will not be correct!
Try one yourself!
Factor 2y+6y2-3y-1 (remember what we just did)
Answer: (2y-1)(3y+1)
2y and 6y2 have the highest common factor of 2y
-3y-1 have the highest common factor of -1
2y+6y2 = 2y(1+3y)
-3y-1 = -(3y+1)
Since both expressions have a (3y+1), we can combine the two (2y with -1 and the two (3y+1)’s)
It’s okay if you’re confused right now. Factoring quadratics is all about practice and getting used to these types of questions!
That being said, please ask us any questions if you have them about quadratics or factoring. We want to make sure that everyone understands the material and we won’t know that unless people ask us for help.