The Discriminant
Check for Understanding –
Given a quadratic equation use the
discriminant to determine the nature
of the roots.
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Given a quadratic equation use the discriminant to determine the nature of the roots. ... The quadratic formula allows you to solve. ANY quadratic equation ...
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2021/2022
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The Discriminant
Check for Understanding – Given a quadratic equation use the discriminant to determine the nature of the roots.
Terms we need to know:
Quadratic Formula
Real number system
Rational numbers
Irrational numbers
Perfect squares
Complex numbers
Imaginary numbers
Let’s Practice
- Record your answer for each
Complex Number System Reals
Rationals (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …)
Whole (0, 1, 2, …) Natural (1, 2, …)
Irrationals (no fractions) pi, e
Imaginary i, 2i, -3-7i, etc.
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
What is the discriminant?
The discriminant is the expression b^2 – 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.
Let’s put all of that information in a chart.
Value of Discriminant (^) Number of RootsType and of Related Function^ Sample Graph
D > 0, D is a perfect square D > 0, D NOT a perfect square
D = 0
D < 0
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 b^2 – 4ac under a square root and what that means in relation to the roots of the equation.
Let’s evaluate the first equation.
x^2 – 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.
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.
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.
Now for the third equation.
x^2 – 10x + 25 = 0
What number is under the radical when simplified? 0
What are the solutions of the equation?
5 (double root)
Last but not least, the fourth equation.
4x^2 – 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. 9 31 8
i
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.