Download The Discriminant

The Discriminant
Check for Understanding –
Given a quadratic equation use the
discriminant to determine the nature
of the roots.
Terms we need to know:
1)
2)
3)
4)
5)
6)
7)
Quadratic Formula
Real number system
Rational numbers
Irrational numbers
Perfect squares
Complex numbers
Imaginary numbers
Examples of Rational and Irrational
numbers
Rational
Irrational
a)2
a) 24
b) 𝜋
3
c) 10
12
b) 24
c) 25
3
d) 8
Most common irrational numbers
involve the √ symbol.
Many square roots, cube roots, etc.
are irrational.
Let’s Practice
• Record your answer for each
Complex Number System
Reals
Imaginary
i, 2i, -3-7i, etc.
Rationals
(fractions, decimals)
Integers
(…, -1, -2, 0, 1, 2, …)
Whole
(0, 1, 2, …)
Natural
(1, 2, …)
Irrationals
(no fractions)
pi, e
THE QUADRATIC FORMULA
1. When you solve using completing the
square on the general formula
2
you get:
b  b 2  4ac
x
2a
ax  bx  c  0
2. This is the quadratic formula!
3. Just identify a, b, and c then substitute
into the formula.
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
2
b – 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.
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.
Let’s put all of that information in a chart.
Value of Discriminant
D > 0,
D is a perfect square
D > 0,
D NOT a perfect square
D=0
D<0
Type and
Number of Roots
Sample Graph
of Related Function
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
2
About the value of b – 4ac under a square root
and what that means in relation to the roots of
the equation.
Solve These…
Use the quadratic formula to solve each
of the following equations?
2
1. x – 5x – 14 = 0
2
2. 2x + x – 5 = 0
2
3. x – 10x + 25 = 0
2
4. 4x – 9x + 7 = 0
Let’s evaluate the first equation.
2
x – 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.
Let’s look at the second equation.
2
2x + 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
1  41
4
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.
2
x – 10x + 25 = 0
What number is under the radical when
simplified?
0
What are the solutions of the equation?
5 (double root)
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.
Last but not least, the fourth equation.
2
4x – 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  i 31
8
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.
Let’s put all of that information in a chart.
Value of Discriminant
Type and
Number of Roots
D > 0,
D is a perfect square
2 real,
rational roots
D > 0,
D NOT a perfect square
2 real,
Irrational roots
D=0
1 real, rational root
(double root)
D<0
2 complex roots
(complex
conjugates)
Sample Graph
of Related Function
Try These.
For each of the following quadratic equations,
a) Find the value of the discriminant, and
b) Describe the number and type of roots.
1. x2 + 14x + 49 = 0
3. 3x2 + 8x + 11 = 0
2. x2 + 5x – 2 = 0
4. x2 + 5x – 24 = 0
The Answers
1. x2 + 14x + 49 = 0
D=0
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
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.
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
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.
2 x 2  7 x  11  0
Discriminant =
a=2, b=7, c=-11
b 2  4ac
(7) 2  4(2)(11)
49  88
Discriminant = 137
Value of discriminant=137
Positive-NON perfect
square
Nature of the Roots –
2 Reals - Irrational
Example #1- continued
Solve using the Quadratic Formula
2 x 2  7 x  11  0
a  2, b  7, c  11
b  b 2  4ac
2a
7  7 2  4(2)(11)
2(2)
7  137
4
2 Reals - Irrational
Solving Quadratic Equations
by the Quadratic Formula
Try the following examples. Do your work on your paper and then check
your answers.
1. x  2 x  63  0
2
2. x  8 x  84  0
2
3. x  5 x  24  0
2
4. x  7 x  13  0
2
5. 3 x 2 5 x  6  0
1.  9, 7 
2.(6, 14)
3.  3,8 
 7  i 3 
4. 

2


 5  i 47 
5. 

6

