Study 1.5 CVC # 1 7 # 1,5,9,13, Class Notes: Prof. G. Battaly, Westchester Community College, NY
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1 GOALS: Find Solutions to quadratic equations by: 1. using Zero Product Principle 2. using the Square Root Property 3. Completing the Square 4. applying the Quadratic Formula Study 1.5 CVC # 1 7 # 1,5,9,13, Link to Complete Square Link to Quadratic Formula General Form for Quadratic Equation in 1 Variable: ax 2 + bx + c = 0 where a, b, c are real #s, a 0 Find Solution to quadratic equation by: 1. factoring & Zero Product Principle 2. u 2 & Square Root Property 3. Completing the Square 4. Quadratic Formula Skip to Quadratic Formula G. Battaly
2 To solve a quadratic equation in factored form: u v = 0 u = 0 v = 0 can solve u v = 8 u =? v =? can't solve: infinitely many u u = 9 u 2 = 9 u = ±3 can solve u v = 0 u = 0 v = 0 can solve To solve a quadratic equation in factored form: u v = 8 u =? v =? can't solve: infinitely many (1)(8) = 8 ( 1)( 8) = 8 (2)(4) = 8 ( 2)( 4) = 8 (1/2)(16) = 8 ( 1/2)( 16) = 8 (1/3)(24) = 8 ( 1/3)( 24) = u u = 9 u 2 = 9 u = ±3 can solve ( 1 ) ( 8 ) G. Battaly
3 x 2 16 = 0 b=0 Can use different approaches to solving. Factoring (x + 4)(x 4) = 0 x + 4 = 0 x 4 = 0 x = 4 x = 4 Find Square Roots x 2 = 16 x = ± 16 x = ± 4 x = 4 and x = 4 are both solutions. x 2 16 = 0 Can use different approaches to solving. Factoring (x + 4)(x 4) = 0 x + 4 = 0 x 4 = 0 x = 4 x = 4 Find Square Roots x 2 = 16 x = ± 16 x = ± 4 x = 4 and x = 4 are both solutions. G. Battaly
4 x 2 19 = 0 b=0 x 2 19 = 0 x 2 = 19 x = ± 19 x = 19 and x = 19 are both solutions. G. Battaly
5 (x 3) 2 = 36 b 0, but left member is a perfect square (x 3) 2 = 36 b 0, but left member is a perfect square Square root property Expand, > general form, factor, solve G. Battaly
6 (x 3) 2 = 36 x 3 = ± 6 x = 3 ± 6 x = 3, 9 3(x + 4) 2 = 21 b 0, but left member is almost a perfect square G. Battaly
7 3(x + 4) 2 = 21 (x + 4) 2 = 7 x + 4 = ± 7 x = 4 ± 7 b 0, but left member is a perfect square (x + 2) 2 = 7 G. Battaly
8 (x + 2) 2 = 7 x + 2 = ± 7 x = 2 ± i 7 x 2 9 = 8 b=0 G. Battaly
9 x 2 9 = 8 b=0 x 2 + 5x + 6 = 0 b 0 Not a perfect square trinomial. ( ) ( ) = 6 ( ) + ( ) = 5?? ( 2 ) ( 3 ) = 6 ( 2 ) + ( 3 ) = 5 Can factor and use Zero Product Principal (ZPP) (x + 2)(x + 3) = 0 x + 2 = 0 x + 3 = 0 x = 2 x = 3 G. Battaly
10 x 2 + 6x + 9 = 0 b 0 x x = 2 (3) x Perfect square trinomial. Use square root property. (x + 3) 2 = 0 x + 3 = ± 0 = 0 x = 3 x 2 + 8x + 16 = 0 b 0 G. Battaly
11 x 2 + 8x + 16 = 0 (x + 4) 2 = 0 x + 4 = 0 x = 4 b 0 x 2 + 6x = 9 b 0 G. Battaly
12 x 2 + 6x = 9 b 0 Write in standard form. x 2 + 6x + 9 = 0 Same as previous. (x + 3) 2 = 0 x + 3 = 0 x = 3 x 2 + 6x + 7 = 0 b 0 Not a Perfect square trinomial. Not factorable w. integers. Can rewrite as a perfect square trinomial? x 2 + 6x = 7 x 2 + 6x + = 7 + APE What can we add to make the left member a perfect square trinomial? x 2 + 6x + 9 = = 2 APE G. Battaly
13 x 2 + 6x + 7 = 0 x 2 + 6x + 9 = = 2 (x + 3) 2 = 2 x + 3 = ± 2 x = 3 ± 2 b 0 x 2 + 6x + 5 = 0 Can we solve using the square root property? Need a perfect square. > Create one by APE. x 2 + 6x = 5 x 2 + 6x + = 5 + APE What can we add to make the left member a perfect square trinomial? b 2 Add to get: b = 2 = 3 2 = 9 Note: coefficient on x 2 term = 1 G. Battaly
14 x 2 + 6x + 5 = 0 x 2 + 6x + 9 = = 4 (x + 3) 2 = 4 x + 3 = ± 2 x = 3 ± 2 x = 5, 1 Was factorable, but this might be easier for problems that are more difficult to factor. b= 2c 0 b = 2 = 3 2 = 9 x x 11= 0 Can we solve using completing the square? Need a perfect square. > Create one by APE. x x = 11 x x + = 11 + What can we add to make the left member a perfect square trinomial? b 2 Add to get: b = 2 = 5 2 =25 Note: coefficient on x 2 term = 1 G. Battaly
15 x x 11 = 0 x x + 25 = (x + 5) 2 = 36 x + 5 = ± 6 x = 5 ± 6 x = 11, 1 Was factorable, but this might be easier. b 0 b 2 2 = = 5 2 =25 2x x 12= 0 Can we solve using completing the square? 1st need a=1. x x 6= 0 Need a perfect square. > Create one by APE. x x = 6 x x + = 6 + What can we add to make the left member a perfect square trinomial? b 2 Add to get: b = 2 = 5 2 =25 Note: coefficient on x 2 term = 1 G. Battaly
16 2x x 12= 0 b 0 x x + 25 = (x + 5) 2 = 31 x + 5 = ± 31 b 2 2 = = 5 2 =25 x = 5 ± 31 2x 2 + 5x 3= 0 x = 3, 1/2 G. Battaly
17 2x 2 + 5x 3= 0 2x x 3 = x x 3 = x x = x x + 25 = ( x + 5 ) 2 = 4 x + 5 = ± 4 x = 5 ± x = = MPE: to get (1)x = ± = 1, 2 3 = 2, 4 or: ( 5+7)/4 = 2/4 = 1/2 ( 5 7)/4 = 12/4 = 3 b = 5/2 b/2 = 5/4 (b/2) 2 = (5/4) 2 = 25/16 G. Battaly
18 x x 20 = 0 Use APE with b 2 2 u 2 = n u = ± n G. Battaly
19 x x = 20 u 2 = n u = ± n x x + = 20 + ( x + ) 2 = 20 + x + = ± 20 + x = ± (80/4) + (121/4) x = ± (201/4) = 11 ± ax 2 + bx + c = 0 Quadratic Formula x = b ± b 2 4ac 2a ALWAYS: 1. Write formula. 2. Write values for a, b, and c. 3. Be careful with substitution, esp. negatives. x x 20 = 0 a = 1, b = 11, c = 20 x = 11 ± (1)( 20) = 11 ± (1) 2 x = 11 ± G. Battaly
20 ax 2 + bx + c = 0 Quadratic Formula x = b ± b 2 4ac 2a ALWAYS: 1. Write formula. 2. Write values for a, b, and c. 3. Be careful with substitution, esp. negatives. 1. x x 11 = x 2 + 5x 3 = x 2 4x 1 = 0 ax 2 + bx + c = 0 x x 11 = 0 Quadratic Formula x = b ± b 2 4ac 2a ALWAYS: 1. Write formula. 2. Write values for a, b, and c. 3. Be careful with substitution, esp. negatives. G. Battaly
21 ax 2 + bx + c = 0 x x 11 = 0 Quadratic Formula x = b ± b 2 4ac 2a ALWAYS: 1. Write formula. 2. Write values for a, b, and c. 3. Be careful with substitution, esp. negatives. 2. 2x 2 + 5x 3 = 0 ax 2 + bx + c = 0 ALWAYS: 1. Write formula. 2. Write values for a, b, and c. 3. Be careful with substitution, esp. negatives. Quadratic Formula x = b ± b 2 4ac 2a G. Battaly
22 2. 2x 2 + 5x 3 = 0 ax 2 + bx + c = 0 ALWAYS: 1. Write formula. 2. Write values for a, b, and c. 3. Be careful with substitution, esp. negatives. Quadratic Formula x = b ± b 2 4ac 2a ax 2 + bx + c = x 2 4x 1 = 0 ALWAYS: 1. Write formula. 2. Write values for a, b, and c. 3. Be careful with substitution, esp. negatives. Quadratic Formula x = b ± b 2 4ac 2a G. Battaly
23 ax 2 + bx + c = x 2 4x 1 = 0 ALWAYS: 1. Write formula. 2. Write values for a, b, and c. 3. Be careful with substitution, esp. negatives. Quadratic Formula x = b ± b 2 4ac 2a ax 2 + bx + c = 0 Derivation of the Quadratic Formula: not required Quadratic Formula G. Battaly
24 Practice 1. 5x 2 20x = x 2 7x + 3 = x 2 = x = 0, 4 2. x = 1/2, 3 3. x = ± 5 5 G. Battaly
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