Factoring (Part 1 )
On the 7th day of quadratics,
my teacher gave to me…
Learning Goals
By the end of the lesson I will be able to:
Review:
The standard form of a quadratic equation is:
y = ax2 + bx + c
From this equation, we can immediately find:
The y-intercept. It’s the c value.
Review:
The factored form of a quadratic equation is:
y = a(x - r)(x - s)
From this equation, we can immediately find:
Review: Algebra Tiles
Expand: (x + 4)(x - 2)
(x - 2)
(x + 4)
+
+
or x2 + 2x - 8
x2
x
-1
1
x
x
-1
-x
-x
-1
-1
x2
1
1
1
x
x
x
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
x
x
x
x
-x
-x
What we know
So we know 3 ways to EXPAND a factored form equation into standard form:
But we don’t know how to convert from standard form into factored form…
yet
Factoring!
Today we’re going to learn how to factor. Factoring is how we convert from standard form to factored form.
We are going to explore 2 techniques:
Algebra Tiles
Anything we can do with algebra tiles, we can undo.
In this video we see the general idea of factoring using algebra tiles:
Creating rectangles
A Hiccup in the process
What if we have a negative? Can we still use algebra tiles?
Yes you can.
We just need to use
zero-pairs: If we add a
positive, we must also
add a negative
Factoring with Algebra Tiles
Factor: y = x2 + 2x + 1
So we have:
+
+
(x + 1)
(x + 1)
Therefore: y = x2 + 2x + 1 = (x + 1)(x + 1)
x2
x
1
x
x2
x
x
1
x
1
x
1
Factoring with Algebra Tiles
Factor: y = x2 - 4x + 3
So we have:
+
+
(x - 3)
(x - 1)
Therefore: y = x2 - 4x - 3 = (x - 3)(x - 1)
x2
-1
1
-x
-x
x2
1
x
x
1
1
-x
-x
-x
1
1
-x
-x
-x
-1
-1
-1
Factoring with Algebra Tiles
Factor: y = x2 + 4x - 12
So we have:
+
+
(x + 6)
(x - 2)
Therefore: y = x2 + 4x - 12 = (x + 6)(x - 2)
x2
-x
1
-1
x
-x
x2
x
x
x
1
x
-1
x
x
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
x
x
x
x
x
1
1
1
1
-1
Sum-Product Method
Let’s take a look at the other method, called the sum-product method.
Remember:
Yesterday we figure out that if we have y = a(x - s)(x - r) we can find the y-intercept if we multiply s by r.
We also established that in y = ax2 + bx + c, c is the y-intercept
More patterns!
Let’s talk about b in y = ax2 + bx + c
Standard Form | Factored Form | b | c | r | s |
y = x2 + 2x + 1 | y = (x + 1)(x + 1) | 2 | 1 | -1 | -1 |
y = x2 - 4x + 3 | y = (x - 3)(x - 1) | -4 | 3 | 3 | 1 |
y = x2 + 4x - 12 | y = (x + 6)(x - 2) | 4 | -12 | -6 | 2 |
y = x2 - x - 12 | y = (x - 4)(x + 3) | | | | |
y = x2 + 6x + 5 | y = (x + 5)(x + 1) | | | | |
y = x2 - 8x + 15 | y = (x - 5)(x - 3) | | | | |
y = x2 - x - 6 | y = (x - 3)(x + 2) | | | | |
More Patterns!
So now we know:
Therefore, we can factor y = x2 + bx + c by finding 2 integers that:
Example
Factor y = x2 + 5x + 4
We need 2 integers that:
What are the factors of 4?
1 x 4 = 4
2 x 2 = 4
-1 x -4 = 4
-2 x -2 = 4
Sum
5
4
-5
-4
So, our factored form is y = (x + 1)(x + 4)
Example
Factor y = x2 + 17x + 30
We need 2 integers that:
What are the
factors of 30? Sum
1 x 30 = 30 31
2 x 15 = 30 17
So, our factored form is y = (x + 2)(x + 15)
Example
Factor y = x2 + 8x - 20
We need 2 integers that:
What are the
factors of -20? Sum
1 x -20 -19
-1 x 20 19
2 x -10 -8
-2 x 10 8
So, our factored form is y = (x - 2)(x + 10)
Example
Factor y = x2 + 7x - 18
We need 2 integers that:
What are the
factors of -18? Sum
1 x -18 -17
-1 x 18 17
2 x -9 -7
-2 x 9 7
So, our factored form is y = (x - 2)(x + 9)
Example
Factor y = x2 - 2x - 24
We need 2 integers that:
What are the
factors of -24? Sum
1 x -24 -23
2 x -12 -10
3 x -8 -5
4 x -6 -2
So, our factored form is y = (x + 4)(x - 6)
Example
Factor y = x2 - 9x + 18
We need 2 integers that:
What are the
factors of 18? Sum
-1 x -18 -19
-2 x -9 -11
-3 x -6 -9
So, our factored form is y = (x - 3)(x - 6)