Properties of Quadratics

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Properties of Quadratics
Determining the Nature of the Roots
We now need to determine the nature of the roots.
In other words, describe what kind of roots we
have. To do that, we need to use part of the
quadratic formula
We are going to use the part of the quadratic
formula under the radical. This is known as the
DISCRIMINANT.
We know that the roots are found where the
quadratic equation intersects the x-axis.
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Using the discriminant

• There are three things we need to check for
• Real vs. Imaginary
• Equal vs. Unequal
• Rational vs. Irrational

• Case 2 Discriminant is zero
• This means that the roots are real, equal and
  rational.

Lets check the four cases
EX Suppose the formula gave us the following

• Case 1 Discriminant is negative
• This means that the roots are imaginary because
  we have a negative under the radical.

So we see that both roots are ½, so they are
equal, and ½ is a real, rational number!
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• Case 4 Discriminant is positive and not a
  perfect square
• This means that the roots are real, unequal and
  irrational.

• Case 3 Discriminant is positive and a perfect
  square
• This means that the roots are real, unequal and
  rational.

EX Suppose the formula gave us the following
EX Suppose the formula gave us the following
So we see that both roots real, irrational
numbers that are NOT equal
So we see that both roots real, rational numbers
that are NOT equal
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Summary 1. Discriminant is negative so roots
are imaginary. ONLY TIME THAT THE ROOTS ARE
IMAGINARY!
3. Discriminant is a positive perfect square so
roots are real, rational and unequal.
2. Discriminant is zero so roots are real,
rational and equal. ONLY TIME THAT THE ROOTS ARE
EQUAL!
4. Discriminant is a positive non-perfect
square so roots are real, irrational and unequal.
ONLY TIME THE ROOTS ARE IRRATIONAL!
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The discriminant is a positive, non-perfect
square. Therefore the roots are 1. Real 2.
Irrational 3. Unequal
The discriminant is a positive, perfect square.
Therefore the roots are 1. Real 2. Rational 3.
Unequal
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The discriminant is zero. Therefore the roots
are 1. Real 2. Rational 3. Equal
The discriminant is negative. Therefore the
roots are 1. Imaginary
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The discriminant is a positive, perfect square.
Therefore the roots are 1. Real 2. Rational 3.
Unequal
The discriminant is a positive, non-perfect
square. Therefore the roots are 1. Real 2.
Irrational 3. Unequal
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Homework

• Page 1
• 3,6,9,11,13,15,17,19,21,24,27,30,33