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

5z(z – 4)

ANSWER

Factor with special patternsFactor the expression.a. 9x2 – 64

= (3x + 8) (3x – 8)

Difference of two squares

b. 4y2 + 20y + 25

= (2y + 5)2

Perfect square trinomial

c. 36w2 – 12w + 1= (6w – 1)2

= (3x)2 – 82

= (2y)2 + 2(2y) (5) + 52

= (6w)2 – 2(6w) (1) + (1)2

Perfect 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 5x2 – 17x + 6.1. 5x2 −17x+6

2. 5x2 −?x −?x+6

3.

5x2 −15x

−2x +6

5x

−2

x −3

ANSWER

Example: Factor 3x2 −17x+10

1. 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)

1. Factors of (3)(10) that add to −17

2. Factor by grouping

3. Rewrite equation

4. Use reverse distributive

5. Answer

Example: Factor 3x2 −17x+10

1. 3x2 −17x+10

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.

3x2 −15x

−2x +10

3x

−2

x −5

Guided PracticeFactor the expression. If the expression cannot be factored, say so.

7x2 – 20x – 3

ANSWER

Guided Practice4x2 – 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 EquationFinding 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.

Example: Solve.2t2-17t+45=3t-5

2t2-17t+45=3t-5 Set eqn. =02t2-20t+50=0 factor out GCF of 22(t2-10t+25)=0 divide by 2t2-10t+25=0 factor left side(t-5)2=0 set factors =0t-5=0 solve for t+5 +5t=5 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 – 200 = 4x2 + 18x – 100 = 2x2 + 9x – 50 = (2x – 1) (x + 5)2x – 1 = 0 or x + 5 = 0

Multiply using FOIL.Write in standard formDivide each side by 2.Factor.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.

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 ?

SolutionSTEP 1 Define the variables. Let x represent the

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) = $288,000= – 2000(2 – 14) (2 + 10)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?