Transcript 5.2: Solving Quadratic Equations by Factoring

1.4 Solving Quadratic
Equations by Factoring
(p. 25)
Day 1
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
Guided Practice
– 5z2 + 20z
ANSWER
5z(z – 4)
Factor with special patterns
Factor the expression.
a. 9x2 – 64 = (3x)2 – 82
= (3x + 8) (3x – 8)
Difference of
two squares
b. 4y2 + 20y + 25 = (2y)2 + 2(2y) (5) + 52
= (2y + 5)2
c.
Perfect
square
trinomial
36w2 – 12w + 1 = (6w)2 – 2(6w) (1) + (1)2
Perfect
= (6w – 1)2
square
trinomial
How to spot patterns
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.
Factor
1. 5x2 −17x+6
2. 5x2 −?x −?x+6
3.
x
−3
5x
5x2
−15x
−2
−2x
+6
ANSWER
2
5x
– 17x + 6.
Example: Factor 3x2 −17x+10
1. 3x2 −17x+10
2. 3x2 −?x −?x+10
1. Factors of (3)(10) that
add to −17
2. Factor by grouping
3. 3x2 −15x −2x+10
3. Rewrite equation
4. 3x(x−5)−2(x−5)
4. Use reverse
distributive
5. (x−5)(3x−2)
5. Answer
Example: Factor 3x2 −17x+10
1. 3x2 −17x+10
2. 3x2 −?x −?x+10
3.
x
−5
3x
3x2
−15x
−2
−2x
+10
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.
Guided Practice
Factor the expression. If the expression
cannot be factored, say so.
7x2 – 20x – 3
ANSWER
Guided Practice
4x2 – 9x + 2
ANSWER
(4x – 1) (x - 2).
Guided Practice
2w2 + w + 3
ANSWER
2w2 + w + 3 cannot be factored
Assignment
p. 29, 3-12 all,
14-30 even, 31
1.4 Solving Quadratic
Equations by Factoring
(p. 25)
Day 2
What is the difference between factoring
an equation and solving an equation?
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!
Finding the Zeros of an Equation
• The Zeros of an equation are the xintercepts !
• First, change y to a zero.
• Now, solve for x.
• The solutions will be the zeros of the
equation.
Example: Solve.
2t2-17t+45=3t-5
2t2-17t+45=3t-5
2t2-20t+50=0
2(t2-10t+25)=0
t2-10t+25=0
(t-5)2=0
t-5=0
+5 +5
t=5
Set eqn. =0
factor out GCF of 2
divide by 2
factor left side
set factors =0
solve for t
check your solution!
Solve the quadratic equation
3x2 + 10x – 8 = 0
ANSWER
Solve the quadratic equation
ANSWER
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?
Solution
10 = 20 + 18x + 4x2 – 20 Multiply using FOIL.
0 = 4x2 + 18x – 10
Write in standard form
2
0 = 2x + 9x – 5
Divide each side by 2.
0 = (2x – 1) (x + 5)
Factor.
2x – 1 = 0 or x + 5 = 0
Zero product property
x = 1 or x = – 5
Solve for x.
2
Reject the negative value, – 5. The border’s
width should be ½ ft, or 6 in.
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 ?
Solution
Define
the
variables.
Let
x
represent
the
STEP 1
price increase and R(x) represent the
annual revenue.
STEP 2 Write a verbal model. Then write and
simplify a quadratic function.
R(x)
R(x)
=
=
(– 2000x + 28,000) (x + 10)
– 2000(x – 14) (x + 10)
STEP 3 Identify the zeros and find their
average. Find how much each
subscription should cost to maximize
annual revenue.
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.
Assignment
p. 29,
32-48 even, 53-58 all
What is the difference between
factoring an equation and solving
an equation?