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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.
2 4
2
b b acx
a
2 0ax bx c
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Identifying a, b, and c
From the quadratic equation,
ax2+bx+c=0The quadratic term is aThe linear term is bThe constant is c
Example:x2 + 2x – 5 = 0
a = 1b = 2c = -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:
b2 – 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 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
22 7 11 0x x
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 = 2
2
4
(7) 4(2)( 11)
49
137
88
b ac
Discriminant =
Value of discriminant=137
Positive-NON perfect square
Nature of the Roots – 2 Reals - Irrational
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Example #1- continued
22 7 11 0x x
2
2
4
2
7 7 4(2)( 11)
2(
2, 7, 11
7 137 2 Reals - Irrational
4
2)
a b
b ac
a
c
b
Solve using the Quadratic Formula
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Solving Quadratic Equations by the Quadratic Formula
2
2
2
2
2
1. 2 63 0
2. 8 84 0
3. 5 24 0
4. 7 13 0
5. 3 5 6 0
x x
x x
x x
x x
x x
Try the following examples. Do your work on your paper and then check your answers.
1. 9,7
2.(6, 14)
3. 3,8
7 34.
2
5 475.
6
i
i
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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.