11.2 Solving Quadratic Equations Using the Quadratic Formula 1. Solve quadratic equations using the quadratic formula. 2. Use the discriminant to determine the number of real solutions that a quadratic equation has. 3. Find the x- and y-intercepts of a quadratic function. 4. Solve applications using the quadratic formula. Solve: ax 2 bx c 0 Coefficient of squared term is NOT 1. b c x x 0 a a 2 2

2 b b c b2 b 4ac 2 x x a 4a 2 4a 2 a 4a 2 4a 2 1 b b 2 a 2a squared

b 2 b 2 4ac x 2a 4a 2 b x 2 a 2 b 2 4ac 4a 2 b x 2a b 2 4ac

2a b x 2a b 2 4ac 2a b b 2 4ac x 2a Quadratic Formula Quadratic Formula To solve ax2 + bx + c = 0, where a 0, use b b 2 4ac x 2a Solve: 2x 2 7 x 3

2x 2 7 x 3 0 a = 2 , b = 7 , c = 3 b b 2 4ac x 2a 2 x 7 7 4 2 3 2 2 7 49 24 x 4 7 25 x 4 7 5 x 4 7 5 x

4 x 3 7 5 x 4 1 x 2 1 ,3 2 2 real rational solutions Solve: x 2 2 x 5 6 x 2 2 x 11 0 a = 1, b = 2, c = 11. 2 x

2 2 4 1 11 x 1 x 2 1 2 4 44 2 2 48 x 2 16 3 2 2 4 3 21 2 4 3 x 2 2

x 1 2 3 1 2 3, 1 2 3 2 real irrational solutions Solve: x 2 5 4x x 2 4x 5 0 a = 1, b = -4, c = 5 x 4 4 2 21 41 5 4 4

x 2 2 x 1 4 2i 21 x 2 i 2 i, 2 i 2 non-real complex solutions Solve using the quadratic formula. 3 x 2 8 x 2 0 a)

8 2 10 6 b) 8 10 3 c) 4 10 3 d) 2 22 3 11.2 Copyright 2011 Pearson Education, Inc. Slide 11- 7 Solve using the quadratic formula. 3 x 2 8 x 2 0 a) 8 2 10 6 b) 8 10

3 c) 4 10 3 d) 2 22 3 11.2 Copyright 2011 Pearson Education, Inc. Slide 11- 8 Methods for Solving Quadratic Equations Method When the Method is Beneficial 1. Factoring Use when the quadratic equation can be easily factored. 2. Square root principle Use when the quadratic equation

can be easily written in the form ax 2 c, or (ax b)2 c. No middle term. 3. Completing the square Rarely the best method, but important for other topics. 4. Quadratic formula Use when factoring is not easy, or possible. 2x 2 7 x 3 0 x 2 2 x 11 0 7 25 x 4 2 48 x 2

1 ,3 2 2 real rational solutions 1 2 x 2 4x 5 0 x 2 i, 3, 1 2 3 2 i 2 non-real complex solutions 2 real irrational solutions

What made the difference? 4 4 2 The Discriminant 2 b 4ac Discriminant: The discriminant is the radicand, b2 4ac, in the quadratic formula. The discriminant is used to determine the number and type of solutions to a quadratic equation. 2x 2 7 x 3 0 x 2 2 x 11 0 x 2 4x 5 0 7 25 x 4

2 48 x 2 4 4 x 2 1 ,3 2 2 real rational solutions If the discriminant is. 1 2 3, 1 2 3 2 real irrational solutions 2 i, 2 i

2 non-real complex solutions there will be. positive and a perfect square 2 real rational solutions. There will be no radicals left in the answer. The equation could have been factored. positive but not a perfect square 2 real irrational solutions. There will be a radical in the answer. 0 1 real rational solution. negative 2 non-real complex solutions. The answer will contain an imaginary number.

Use the discriminant to determine the number and type of solutions. 2 x 2 5 x 1 2 x 2 5 x 1 0 Evaluate the discriminant: b2 4ac. a = 2, b = 5, c = 1 5 2 4 2 1 25 8 Discriminant: 17 Positive but not a perfect square. Two real irrational solutions. Find the discriminant. x 2 7 x 6 a) 5 b) 73

c) 25 d) 11.2 73 Copyright 2011 Pearson Education, Inc. Slide 11- 14 Find the discriminant. x 2 7 x 6 a) 5 b) 73 c) 25 d) 11.2 73 Copyright 2011 Pearson Education, Inc. Slide 11- 15 Determine the number and type of solutions. x 2 7 x 6

a) Two real rational solutions b) Two real irrational solutions c) One real rational solution. d) Two non-real complex solutions. 11.2 Copyright 2011 Pearson Education, Inc. Slide 11- 16 Determine the number and type of solutions. x 2 7 x 6 a) Two real rational solutions b) Two real irrational solutions c) One real rational solution. d) Two non-real complex solutions. 11.2 Copyright 2011 Pearson Education, Inc. Slide 11- 17 Find the discriminant. 2 x 2 4 x 15

a) 136 b) 136 c) -104 d) 104 11.2 Copyright 2011 Pearson Education, Inc. Slide 11- 18 Find the discriminant. 2 x 2 4 x 15 a) 136 b) 136 c) -104 d) 11.2

104 Copyright 2011 Pearson Education, Inc. Slide 11- 19 Determine the number and type of solutions. 2 2 x 4 x 15 a) Two real rational solutions b) Two real irrational solutions c) One real rational solution. d) Two non-real complex solutions. 11.2 Copyright 2011 Pearson Education, Inc. Slide 11- 20 Determine the number and type of solutions. 2 2 x 4 x 15 a) Two real rational solutions

b) Two real irrational solutions c) One real rational solution. d) Two non-real complex solutions. 11.2 Copyright 2011 Pearson Education, Inc. Slide 11- 21 f x x 2 2 f x x x 12 y x 2 x 12 0 x 2 x 12 What are we finding? x-intercepts 0 x 4 x 3 x 4 x 3

4, 0 3, 0 y-intercept 2 f 0 0 0 12 12 0, 12

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