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1BMath SL Year 2
Name:
Date:
1-7 Factoring Quadratic
K
Notes
1.
•
What do I need to know?
Definitions for:
o Factor/Factorize
o Quadratic
o Roots/Zeros
o Discriminant
o Quadratic Equation
o Axis of Symmetry
Notes to Self
o Real Roots
o No solution
o Unique/distinct/differentsolutions
•
Notation for:
o ax2+ bc+c
• Processes:
o
o
o
o
Quadratic formula
Using the discriminant
Factorizing a quadratic
Finding the axis of symmet
In this lesson we will revisit the following learning goals:
1. How do we factor a trinomial with a lead coefficient of 1?
2. How do we solve a quadratic equation if it is not factorable?
3. How do we calculate the discriminant of a quadratic?
3. What does the discriminant tell us about the roots?
Fact Check!
•
A quadratic function is an equation in the form
•
We can also write this as f: x
•
Solving quadratic equations:
o Factor (factorise) —the null factor law
• If ab=O, then a = O or b
o Complete the square
o Quadratic formula
o Technology (GDC)
The
o
ax2+ bx + c,
where a, b, care constants, a
O
can be used when a quadratic is not factorable
Nature of the roots can be determined.
O
1BMath SL Year 2
Factoring = Factorizing
Method of Factorin
GCF
Process
1. Determine the largest factor all
terms have in common
2. Divide that factor out of each
•2x -4x
Factor:
-36<+2)
term
Difference of Two
Trvl
Example
Use when two perfect squares are
subtracted
Factor: 9 —t
act •r -16
1. Set = O.
2. Determine which factors of
c add to b
3. Insert that pair into binomial
factors
Solve:
olve:
2
Squares
Sum and Product
Y = -1 and y = 3
Example: 3x2—11x—4
1. Multiply a and c
2. Determine two factors of that
-12x1=-
product (ac) that add to b
3. Replace the middle term with
those two numbers
4. Factor by grouping
Factor:
12
-12+1=-11
Rewrite: 3x2—12x+ Ix—4
GCF = 3x
AC Method
3x (x—4)
GCF
+1 (x—
4)
Factors: (3x + I) (x —4)
I. Identify a, b, and c
2. Use formula provided in booklet
Quadratic Formula
cdð
Math SL Year 2
The Discriminant
The Quadratic Formula- We use this to SOLVE quadratic equations that are not factorable
Let's find it in our formula booklet:
Solutions of a quadratic
equation
In the quadratic formula, the quantity b2 —'lac under the square root sign is called the
discriminant.
The symbol delta A is used to represent the discriminant, so A
The quadratic formula becomes
•
If A
—
l/ —
where A replaces b2 —'Inc.
0, there is
• If > O,thereare
•
If b < 0, there are
•
there are
If a, b, c, are rational and A is a perfect square,
Summary Table:
coaOð bq
Discriminant value
mots of quadratic
two real distinct roots
two identical real roots (repeated)
no real roots
of 2x a
tumple) use the discriminant to determine the naturo of tho roots
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Math SL Year 2
Solve each equation using the quadratic formula. Where necessary reduce the radical to simplest form.
11. x 2 +4x-6
0
12. 2x 2 —
32
...IL
13. 3x 2 = 7x +6
axq-q-
q6X-Cò
20)
(-2
14. Use the discriminant to determine the nature of the roots of each:
a. 9x 2 +6x+1=o
b. 3x — 5
4
x
-xS +18
IS. For x a
2x + m
0, find A and ence
root V -yac
a. A repeated
1.1
b. 2 distinct real roots
C.
toots
a ues of m for which the equation has:
z C)
3
Co
O
ath SL Year 2
16. For the equation kx2 + (k + 3)x
a repeated root .
find the discriminant and determine the value of k for which the equation has
-z- discrìmtncqxt
o
17. Find the value(s) of k for which the equation 2x 2 — kx + 3 = 0 wt
ave
F-qaC > O
18. Find the values of p such that the equation has two different real roots.
a. px2+5x+2=0
Sž-qp2
26 —
0
- 8p7 -as
b. x2+3px+1=0
bZ-HQC
? C)
rent real roots.