1.4 Solving Quadratic Equations by Factoring

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1.4 Solving Quadratic Equations by Factoring

• (p. 25)
• Day 1

 
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(No Transcript)
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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
 
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Guided Practice

• 5z2 20z

ANSWER
5z(z 4)
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Factor with special patterns
Factor the expression.
Difference of two squares
a. 9x2 64
(3x)2 82
(3x 8) (3x 8)
b. 4y2 20y 25
Perfect square trinomial
(2y)2 2(2y) (5) 52
(2y 5)2
(6w)2 2(6w) (1) (1)2
c. 36w2 12w 1
Perfect square trinomial
(6w 1)2
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How to spot patterns

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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 gt 0, m and n have the same
sign. So, m and n must both be negative because
the coefficient of x, 17, is negative.
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Factor 5x2 17x 6.

• 5x2 -17x6
• 2. 5x2 -?x -?x6
• 3.

 
x
-3
5x2
-15x
5x
-2
-2x
6
 
ANSWER
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Example Factor 3x2 -17x10
1. Factors of (3)(10) that add to -17 2. Factor
by grouping 3. Rewrite equation 4. Use reverse
distributive 5. Answer

• 3x2 -17x10
• 2. 3x2 -?x -?x10
• 3. 3x2 -15x -2x10
• 4. 3x(x-5)-2(x-5)
• 5. (x-5)(3x-2)

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Example Factor 3x2 -17x10

• 3x2 -17x10
• 2. 3x2 -?x -?x10
• 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
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Guided Practice
Factor the expression. If the expression cannot
be factored, say so.
7x2 20x 3
ANSWER
 
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Guided Practice
4x2 9x 2
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Guided Practice

• 2w2 w 3

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•  

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Assignment
p. 29, 3-12 all, 14-30 even, 31
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1.4 Solving Quadratic Equations by Factoring
(p. 25) Day 2
What is the difference between factoring an
equation and solving an equation?
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Zero Product Property

• Let A and B be real numbers or algebraic
  expressions. If AB0, then A0 or B0.
• This means that If the product of 2 factors is
  zero, then at least one of the 2 factors had to
  be zero itself!

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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.

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Example Solve.2t2-17t453t-5

• 2t2-17t453t-5 Set eqn. 0
• 2t2-20t500 factor out GCF of 2
• 2(t2-10t25)0 divide by 2
• t2-10t250 factor left side
• (t-5)20 set factors 0
• t-50 solve for t
• 5 5
• t5 check your solution!

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Solve the quadratic equation

• 3x2 10x 8 0

ANSWER
 
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Solve the quadratic equation

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ANSWER
 
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Use a quadratic equation as a model
Quilts
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Solution
10 20 18x 4x2 20
Multiply using FOIL.
0 4x2 18x 10
Write in standard form
0 2x2 9x 5
Divide each side by 2.
0 (2x 1) (x 5)
Factor.
Zero product property
Solve for x.
Reject the negative value, 5. The borders
width should be ½ ft, or 6 in.
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Magazines
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Solution
Define the variables. Let x represent the price
increase and R(x) represent the annual revenue.
STEP 1
Write a verbal model. Then write and simplify a
quadratic function.
STEP 2
R(x)
( 2000x 28,000) (x 10)
R(x)
2000(x 14) (x 10)
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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

2.
To maximize revenue, each subscription should
cost 10 2 12.
STEP 4
Find the maximum annual revenue.
R(2)
288,000
2000(2 14) (2 10)
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Assignment
p. 29, 32-48 even, 53-58 all What is the
difference between factoring an equation and
solving an equation?