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Notes Packet 10: Solving Quadratic Equations by the Quadratic Formula

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THE QUADRATIC FORMULA 1.The general quadratic equation 2.This is the quadratic formula! 3.Just identify a, b, and c then substitute into the formula.

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Identifying a, b, and c From the quadratic equation, ax 2 +bx+c=0 The quadratic term is a The linear term is b The constant is c Example: x 2 + 2x – 5 = 0 a = 1 b = 2 c = -5

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WHY USE THE QUADRATIC FORMULA? The quadratic formula allows you to solve ANY quadratic equation, even if you cannot graph 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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Example #1- continued Solve using the Quadratic Formula

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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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Complex Numbers

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Definition of pure imaginary numbers: Any positive real number b, where i is the imaginary unit and bi is called the pure imaginary number.

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Definition of pure imaginary numbers: i is not a variable it is a symbol for a specific number

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Simplify each expression.

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Remember Simplify each expression. Remember

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Simplify. To figure out where we are in the cycle divide the exponent by 4 and look at the remainder.

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Simplify. Divide the exponent by 4 and look at the remainder.

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Simplify. Divide the exponent by 4 and look at the remainder.

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Simplify. Divide the exponent by 4 and look at the remainder.

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