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 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. 36w2 – 12w + 1 = (6w)2 – 2(6w) (1) + (1)2 Perfect square trinomial = (6w – 1)2

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. 5x2 −17x+6 2. 5x2 −?x −?x+6 3. x −3 5x2 −15x 5x   x −3 5x2 −15x 5x −2 −2x +6 ANSWER  

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)

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

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

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 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. =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!

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. 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.

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 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)

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.