1. 1.4 Solving Quadratic Equations by Factoring
(p. 25)
Day 1

3. Factor the Expression
The first thing we should look for and it is the last thing we think about---
Is there any number or variable common to all of the terms?
ANSWER

4. Guided Practice
– 5z2 + 20z
ANSWER
5z(z – 4)

5. Factor with special patterns
Factor the expression.
a. 9x2 – 64
= (3x)2 – 82
Difference of two squares
= (3x + 8) (3x – 8)
b. 4y2 + 20y + 25
= (2y)2 + 2(2y) (5) + 52
Perfect square trinomial
= (2y + 5)2
c w2 – 12w + 1
= (6w)2 – 2(6w) (1) + (1)2
Perfect square trinomial
= (6w – 1)2

6. How to spot patterns

7. Factor 5x2 – 17x + 6.
SOLUTION
You want 5x2 – 17x + 6 = (kx + m) (lx + n) where k and l are factors of 5 and m and n are factors of 6. You can assume that k and l are positive and k ≥ l. Because mn > 0, m and n have the same sign. So, m and n must both be negative because the coefficient of x, – 17, is negative.

8. Factor 5x2 – 17x + 6. 5x2 −17x+6 2. 5x2 −?x −?x+6 3. x −3 5x2 −15x 5xx −3
5x2
−15x 5x −2
−2x +6
ANSWER

9. Example: Factor 3x2 −17x+10 1. Factors of (3)(10) that add to −17
2. Factor by grouping
3. Rewrite equation
4. Use reverse distributive
5. Answer
3x2 −17x+10
2. 3x2 −?x −?x+10
3. 3x2 −15x −2x+10
4. 3x(x−5)−2(x−5)
5. (x−5)(3x−2)

10. Example: Factor 3x2 −17x+10 1.Rewrite the equation
2. 3x2 −?x −?x+10 3.
1.Rewrite the equation
2. Factors of (3)(10) that add to −17 (−15 & −2)
3. Place each term in a box from right to left.
4. Take out common factors in rows.
5. Take out common factors in columns. x −5
3x2
−15x 3x −2
−2x
+10

11. Guided Practice
Factor the expression. If the expression cannot be factored, say so.
7x2 – 20x – 3
ANSWER

12. Guided Practice
4x2 – 9x + 2
ANSWER
(4x – 1) (x - 2).

13. Guided Practice
2w2 + w + 3
ANSWER
2w2 + w + 3 cannot be factored

14. 1.4 Solving Quadratic Equations by Factoring
(p. 25)
Day 2
What is the difference between factoring an equation and solving an equation?

16. Zero Product Property
Let A and B be real numbers or algebraic expressions. If AB=0, then A=0 or B=0.
This means that If the product of 2 factors is zero, then at least one of the 2 factors had to be zero itself!

17. Finding the Zeros of an Equation
The Zeros of an equation are the x-intercepts !
First, change y to a zero.
Now, solve for x.
The solutions will be the zeros of the equation.

18. Example: Solve. 2t2-17t+45=3t-5
2t2-17t+45=3t-5 Set eqn. =0
2t2-20t+50=0 factor out GCF of 2
2(t2-10t+25)=0 divide by 2
t2-10t+25=0 factor left side
(t-5)2=0 set factors =0
t-5=0 solve for t
+5 +5
t=5 check your solution!

19. Solve the quadratic equation
3x2 + 10x – 8 = 0
ANSWER

20. Solve the quadratic equation
ANSWER

21. Use a quadratic equation as a model
Quilts
You have made a rectangular quilt that is 5 feet by 4 feet. You want to use the remaining 10 square feet of fabric to add a decorative border of uniform width to the quilt. What should the width of the quilt’s border be?

22. Solution 10 = 20 + 18x + 4x2 – 20 Multiply using FOIL.
Write in standard form
0 = 2x2 + 9x – 5
Divide each side by 2.
0 = (2x – 1) (x + 5)
Factor.
2x – 1 = 0
or x + 5 = 0
Zero product property 12
x =
or x = – 5
Solve for x.
Reject the negative value, – 5. The border’s width should be ½ ft, or 6 in.

23. Magazines
A monthly teen magazine has 28,000 subscribers when it charges $10 per annual subscription. For each $1 increase in price, the magazine loses about 2000 subscribers. How much should the magazine charge to maximize annual revenue ? What is the maximum annual revenue ?

24. Solution
Define the variables. Let x represent the price increase and R(x) represent the annual revenue.
STEP 1
STEP 2
Write a verbal model. Then write and simplify a quadratic function.
R(x)
= (– 2000x + 28,000) (x + 10)
R(x)
= – 2000(x – 14) (x + 10)

25. To maximize revenue, each subscription should cost $10 + $2 = $12.
Identify the zeros and find their average. Find how much each subscription should cost to maximize annual revenue.
STEP 3
The zeros of the revenue function are 14 and –10. The average of the zeroes is
14 + (– 1 0) 2
= 2.
To maximize revenue, each subscription should cost $10 + $2 = $12.
STEP 4
Find the maximum annual revenue.
R(2)
= – 2000(2 – 14) (2 + 10)
= $288,000
ANSWER
The magazine should charge $12 per subscription to maximize annual revenue. The maximum annual revenue is $288,000.