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**The Discriminant Check for Understanding –**

Given a quadratic equation use the discriminant to determine the nature of the roots.

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**Terms we need to know: Quadratic Formula Real number system**

Rational numbers Irrational numbers Perfect squares Complex numbers Imaginary numbers

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**Examples of Rational and Irrational numbers**

2 12 24 25 3 8 24 𝜋 3 10 Most common irrational numbers involve the √ symbol. Many square roots, cube roots, etc. are irrational.

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Let’s Practice Record your answer for each

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**Complex Number System Reals Rationals (fractions, decimals) Integers**

Imaginary i, 2i, -3-7i, etc. Rationals (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …) Irrationals (no fractions) pi, e Whole (0, 1, 2, …) Natural (1, 2, …)

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THE QUADRATIC FORMULA When you solve using completing the square on the general formula you get: This is the quadratic formula! Just identify a, b, and c then substitute into the formula.

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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

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**What is the discriminant?**

The discriminant is the expression b2 – 4ac. We represent the discriminant with D The value of the discriminant can be used to determine the number and type of roots of a quadratic equation.

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**How have we previously used the discriminant?**

We used the discriminant to determine whether a quadratic polynomial could be factored. If the value of the discriminant for a quadratic polynomial is a perfect square, the polynomial can be factored.

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**Let’s put all of that information in a chart.**

Value of Discriminant Type and Number of Roots Sample Graph of Related Function D > 0, D is a perfect square D NOT a perfect square D = 0 D < 0

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**During this presentation, we will complete a chart**

that shows how the value of the discriminant relates to the number and type of roots of a quadratic equation. Rather than simply memorizing the chart, think About the value of b2 – 4ac under a square root and what that means in relation to the roots of the equation.

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Solve These… Use the quadratic formula to solve each of the following equations? x2 – 5x – 14 = 0 2x2 + x – 5 = 0 x2 – 10x + 25 = 0 4x2 – 9x + 7 = 0

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**Let’s evaluate the first equation.**

x2 – 5x – 14 = 0 What number is under the radical when simplified? 81 The discriminant is 81; which is a perfect square What are the solutions of the equation? –2 and 7 which are both rational numbers.

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**If the value of the discriminant is positive,**

the equation will have 2 real roots. If the value of the discriminant is a perfect square, the roots will be rational.

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**Let’s look at the second equation.**

2x2 + x – 5 = 0 What number is under the radical when simplified? 41 (which is not a perfect square.) What are the solutions of the equation? Both solutions are irrational

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**If the value of the discriminant is positive,**

the equation will have 2 real roots. If the value of the discriminant is a NOT perfect square, the roots will be irrational.

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**Now for the third equation.**

x2 – 10x + 25 = 0 What number is under the radical when simplified? What are the solutions of the equation? 5 (double root)

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**If the value of the discriminant is zero,**

the equation will have 1 real, root; it will be a double root. If the value of the discriminant is 0, the roots will be rational.

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**Last but not least, the fourth equation.**

4x2 – 9x + 7 = 0 What number is under the radical when simplified? –31 What are the solutions of the equation? There are no real solutions. The solution is imaginary.

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**If the value of the discriminant is negative,**

the equation will have 2 complex roots; they will be complex conjugates. Not to panic if you don’t recognize these above terms… we will cover them in the next lesson.

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**Let’s put all of that information in a chart.**

Value of Discriminant Type and Number of Roots Sample Graph of Related Function D > 0, D is a perfect square 2 real, rational roots D NOT a perfect square Irrational roots D = 0 1 real, rational root (double root) D < 0 2 complex roots (complex conjugates)

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**Try These. For each of the following quadratic equations,**

Find the value of the discriminant, and Describe the number and type of roots. x2 + 14x + 49 = x2 + 8x + 11 = 0 2. x2 + 5x – 2 = x2 + 5x – 24 = 0

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**The Answers x2 + 14x + 49 = 0 3. 3x2 + 8x + 11 = 0 D = 0 D = –68**

1 real, rational root (double root) 2. x2 + 5x – 2 = 0 D = 33 2 real, irrational roots 3. 3x2 + 8x + 11 = 0 D = –68 2 complex roots (complex conjugates) 4. x2 + 5x – 24 = 0 D = 121 2 real, rational roots

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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 Discriminant Nature of the Solutions Negative 2 imaginary solutions Zero 1 Real Solution Positive – perfect square 2 Reals- Rational Positive – non-perfect square 2 Reals- Irrational

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**Example #1 a=2, b=7, c=-11 Discriminant = Discriminant =**

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 Discriminant =

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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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