120_13.2_notes

120_13.2_notes - Mat120 Chapter 13 — Section 2 Page 9 The Quadratic Formula Solving equations of the form ax2 bx c = 0 What We have seen so far

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Unformatted text preview: Mat120 Chapter 13 — Section 2 Page 9 The Quadratic Formula Solving equations of the form: ax2+ bx + c = 0. What We have seen so far —— M911: __a__LngF etc i : Example 1: 2x2 — x —— 1 = 0 factor: (2x + 1)(x —— 1) : O solve: 2x+1=0 x-1=O x=—1/2 x=1 Method 2: Square Root Fromm: Examplez: x2 = 16 J)? = :Jfi x = i4 so x=4 and X: —4 Method 3: o ' re: x2+2x—4=0 x2+2x=4 Take(§)=12=1 x2 + 2x +1 2 4+1 (x+1)2 = 5 (x+1)2 = iJ§ x+1 = i 5 x = -1iJ§ Adding a new method Method4: a' or u Mat120 Chapter 13 — Section 2 Development of the Quadratic Formula: Given: ax2 + bx + c = 0 Step 1: Coefl'icient of x2 is not 1 so divide it out: b c x2+—x+—=0 a a Step 2: X's on the left and numbers on the right 2 b C x+—-x=—— a a Step 3: Complete the square on the x's and add the result to both sides 2 2 2 2 1 . . 1 b b b b2 x2+2x+ b2 —-9-+—-—- a Z? a 4a2 Put} the right hand side over a common denominator of 4a2 a 4a2 ' 4a a 4a2 x2+9x+ b2 ——4aC+—-tf— a I? ‘ 4a 4a2 x2+9x+£i -2132 a 4a2 4a2 4a2 2 2 x2+9x+£3 =b ‘24ac a 4a 4a ( b )2 b2 — 4ac 4a2 Page 10 Mat120 Chapter 13 — Section 2 Page 11 The Quadratic Formula Step 4: Use the Square Root Property to Solve bf b2—4ac x+— =i"——-——— 2a 4a2 ( b) b2—4ac x+—- =i 2a 4a2 (x+_t_)_)_ b2—4ac 2a —” 2a X : 3i b2—4ac 2a 2a -bi 2—4ac X z Mat120 Chapter 13 — Section 2 Page 12 In Exercises 1-18, solve each equation using the quadratic formula. Simplify, if possible. 2. xz+8x+15=0 Step 1: Write the equation in standard form: ax2+bx+c=0 Step2: List a,b,andc a: I b: y C: IS‘. Step 3: “1 Place those numbers into the quadratic formula X 2 "" 3 j: (3 )A“ 4(1 Holy} .2c 1 ) Step 4: Simplify the answer A T X” 1515?: XS‘Lg—l-é -e X} a,“ r Step 1: Write the equation in standard form: L ax2+bx+c=0 :0 Step2: Est a,b,andc 61:4, bag C__‘L Step 3: Place those numbers into the quadratic formula a? (9) Step 4: Simplify the answer Mat120 Chapter 13 — Section 2 Page 13 The Quadratic Formula Write the equation in standard form: ax2 + bx + c = 0 List a, b, and c Place those numbers into the quadratic formula 0759) Step 4: Simplify the answer x __ _[ + __ 7 I ’ 18. 22(z + 4) = 31 — 3 Write the equation in standard form: ax2 + bx + c = 0 ‘23:? -t_5 a +3 2' List a, b, and c Step 3: Place those numbers into the r” quadratic formula 2- == ‘5‘ i 5 “ 4a)“) 97-01) Step 4: Simplify the answer ’ i 1, -5'2.‘ \j l L/ _. «5‘ :1: I 2 _. A} _. 52. l s 1....5’ 1” g = 7;“ 4 «.2 :: “g” g j ‘l —- 2 Mat120 Chapter 13 — Section 2 I Page 14 7. The Discriminate: 5 " Vac. It will determine the TYPE or solution the quadratic equation will have. Lookat: V 631-441.— 4» bl—w c 20 .1 «acme/«Am; In Exercises 19 - 30, compute the discriminate. Then determine the number and type of solutions for the given equation. 20. X2+8X+3=0 24. 2x2_4x +3=0 0e: 538’ C33 &:Q. bz—‘f C35 513—4-th— 63%; 2’3“. 4an3) («0540.)0) (pg—m; >0 /é~2</ <0 a ail/Mai- So€ 02’ ’m“‘7’“"’"’\1 26. 3X2—5x=0 30_ 4x2=20x_25 a1?) Ira="$—_C:O :70 621944.; Q :4 1);-M (11,23 sttq(5)(0) 25"—-0 >0 («gatwwaw _ :25 47AM 41M Mat120 Chapter 13 — Section 2 The Quadratic Formula The Zero-Product Principle in Reverse If A=0 or B=0,thenAB=0 Example: Write a quadratic equation with the given solution set: {3,—2} The above says: 2:: 3 or x=—2 x—3=0 or x+2=0 (x—3)(x+2)=0 xzwx—6=0 52 {—2, 6} 54 _. _. c, :: (J .— C” X—I-J —-0 X G X :: —- 5’ -9) :0 , (X421) (X bx+$ =0 /Yx*+9>(-b =0 56 {—8i,81} 58 {— J§,J§} =46 744$ )(343 x::/3 X+Y1130 X‘gl‘zr) )(4-fi :o X”J§:O OHS/x) CX-Q: 3:0 (»<+tf3><><*‘/§):O Page 15 ...
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This note was uploaded on 05/05/2008 for the course MAT 120/222/10 taught by Professor None during the Spring '08 term at ASU.

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120_13.2_notes - Mat120 Chapter 13 — Section 2 Page 9 The Quadratic Formula Solving equations of the form ax2 bx c = 0 What We have seen so far

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