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1.4 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. – 5 z 2 + 20 z. ANSWER. 5 z ( z – 4).

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1.4 Solving Quadratic Equations by Factoring

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

  2. 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

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

  4. 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

  5. How to spot patterns

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

  7. Factor 5x2 – 17x + 6. • 5x2 −17x+6 • 2. 5x2 −?x −?x+6 • 3. x −3 5x2 −15x 5x −2 −2x +6 ANSWER

  8. Example: Factor 3x2 −17x+10 • 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

  9. Example: Factor 3x2 −17x+10 • 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. x −5 3x2 −15x 3x −2 −2x +10

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

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

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

  13. Assignment p. 29, 3-12 all, 14-30 even, 31

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

  15. 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!

  16. 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.

  17. 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!

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

  19. Solve the quadratic equation ANSWER

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

  21. orx + 5 = 0 2x – 1 = 0 12 x = orx = – 5 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 border’s width should be ½ ft, or6 in.

  22. 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 ? Magazines

  23. Solution Define the variables. Let xrepresent 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)

  24. The magazine should charge $12 per subscription to maximize annual revenue. The maximum annual revenue is $288,000. ANSWER 14 + (– 1 0) 2 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) =–2000(2–14) (2+ 10) = $288,000

  25. Assignment p. 29, 32-48 even, 53-58 all What is the difference between factoring an equation and solving an equation?

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