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Objective Solving Quadratic Equations by the Quadratic Formula

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Intro: Intro: We already know the standard form of a quadratic equation is: y = ax2 ax2 ax2 ax2 + bx bx + c The The constants constants are: a, b, c The The variables variables are: y, x

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The ROOTS (or solutions) of a polynomial are its x-intercepts The ROOTS (or solutions) of a polynomial are its x-intercepts Recall: The x- intercepts occur where y = 0. Recall: The x- intercepts occur where y = 0.

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Example: Find the roots: y = x 2 + x - 6 Example: Find the roots: y = x 2 + x - 6 Solution: Factoring: y = (x + 3)(x - 2) Solution: Factoring: y = (x + 3)(x - 2) 0 = (x + 3)(x - 2) 0 = (x + 3)(x - 2) The roots are: The roots are: x = -3; x = 2 x = -3; x = 2

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But what about NASTY trinomials that don’t factor? But what about NASTY trinomials that don’t factor?

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After centuries of work, mathematicians realized that as long as you know the coefficients, you can find the roots of the quadratic. Even if it doesn’t factor! After centuries of work, mathematicians realized that as long as you know the coefficients, you can find the roots of the quadratic. Even if it doesn’t factor!

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Example #1- continued Solve using the Quadratic Formula

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Plug in your answers for x. If you’re right, you’ll get y = 0.

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Remember: All the terms must be on one side BEFORE you use the quadratic formula. Example: Solve 3m 2 - 8 = 10m Example: Solve 3m 2 - 8 = 10m Solution: 3m 2 - 10m - 8 = 0 Solution: 3m 2 - 10m - 8 = 0 a = 3, b = -10, c = -8 a = 3, b = -10, c = -8

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Solve: 3x 2 = 7 - 2x Solve: 3x 2 = 7 - 2x Solution: 3x 2 + 2x - 7 = 0 Solution: 3x 2 + 2x - 7 = 0 a = 3, b = 2, c = -7 a = 3, b = 2, c = -7

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WHY USE THE QUADRATIC FORMULA? The quadratic formula allows you to solve ANY quadratic equation, even if you cannot factor it. An important piece of the quadratic formula is what’s under the radical: b 2 – 4ac This piece is called the discriminant.

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WHY IS THE DISCRIMINANT IMPORTANT? The discriminant tells you the number and types of answers (roots) you will get. The discriminant can be +, –, or 0 which actually tells you a lot! Since the discriminant is under a radical, think about what it means if you have a positive or negative number or 0 under the radical.

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WHAT THE DISCRIMINANT TELLS YOU! Value of the DiscriminantNature of the Solutions Negative2 imaginary solutions Zero1 Real Solution Positive – perfect square2 Reals- Rational Positive – non-perfect square 2 Reals- Irrational

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Example #1 Find the value of the discriminant and describe the nature of the roots (real,imaginary, rational, irrational) of each quadratic equation. Then solve the equation using the quadratic formula) 1. a=2, b=7, c=-11 Discriminant = Value of discriminant=137 Positive-NON perfect square Nature of the Roots – 2 Reals - Irrational

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Solving Quadratic Equations by the Quadratic Formula Try the following examples. Do your work on your paper and then check your answers.

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