Solving Quadratic Equations Using the Quadratic Formula 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.

Coefficient of squared term is NOT 1. Solve: Coefficient of squared term is NOT 1. ½ squared Quadratic Formula

Quadratic Formula To solve ax2 + bx + c = 0, where a 0, use

2 real rational solutions Solve: a = 2 , b = –7 , c = 3 2 real rational solutions

2 real irrational solutions Solve: a = 1, b = –2, c = –11. 2 1 1 16 ∙3 2 real irrational solutions

2 non-real complex solutions Solve: a = 1, b = -4, c = 5 2 1 1 2 non-real complex solutions

Copyright © 2011 Pearson Education, Inc. Solve using the quadratic formula. a) b) c) d) Copyright © 2011 Pearson Education, Inc. 11.2

Copyright © 2011 Pearson Education, Inc. Solve using the quadratic formula. a) b) c) d) Copyright © 2011 Pearson Education, Inc. 11.2

Methods for Solving Quadratic Equations 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 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.

What made the difference? The Discriminant 2 non-real complex solutions 2 real rational solutions 2 real irrational solutions What made the difference? The Discriminant

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.

2 non-real complex solutions 2 real rational solutions 2 real irrational solutions If the discriminant is…. 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. 1 real rational solution. negative 2 non-real complex solutions. The answer will contain an imaginary number.

Evaluate the discriminant: b2 – 4ac. Use the discriminant to determine the number and type of solutions. Evaluate the discriminant: b2 – 4ac. a = 2, b = 5, c = 1 Discriminant: Positive but not a perfect square. Two real irrational solutions.

Copyright © 2011 Pearson Education, Inc. Find the discriminant. a) 5 b) 73 c) 25 d) Copyright © 2011 Pearson Education, Inc. 11.2

Copyright © 2011 Pearson Education, Inc. Find the discriminant. a) 5 b) 73 c) 25 d) Copyright © 2011 Pearson Education, Inc. 11.2

Copyright © 2011 Pearson Education, Inc. Determine the number and type of solutions. a) Two real rational solutions b) Two real irrational solutions c) One real rational solution. d) Two non-real complex solutions. Copyright © 2011 Pearson Education, Inc. 11.2

Copyright © 2011 Pearson Education, Inc. Determine the number and type of solutions. a) Two real rational solutions b) Two real irrational solutions c) One real rational solution. d) Two non-real complex solutions. Copyright © 2011 Pearson Education, Inc. 11.2

Copyright © 2011 Pearson Education, Inc. Find the discriminant. a) b) 136 c) -104 d) Copyright © 2011 Pearson Education, Inc. 11.2

Copyright © 2011 Pearson Education, Inc. Find the discriminant. a) b) 136 c) -104 d) Copyright © 2011 Pearson Education, Inc. 11.2

Copyright © 2011 Pearson Education, Inc. Determine the number and type of solutions. a) Two real rational solutions b) Two real irrational solutions c) One real rational solution. d) Two non-real complex solutions. Copyright © 2011 Pearson Education, Inc. 11.2

Copyright © 2011 Pearson Education, Inc. Determine the number and type of solutions. a) Two real rational solutions b) Two real irrational solutions c) One real rational solution. d) Two non-real complex solutions. Copyright © 2011 Pearson Education, Inc. 11.2

What are we finding? x-intercepts y-intercept