Download Quadratic Formula

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The quadratic formula can be used to determine the maximum height, time in
flight, and velocity of a golf ball. This information is useful to club manufactures,
ball manufactures, and golfers alike.
The Quadratic Formula
The quadratic formula can be used to factor
any quadratic equation.
For a quadratic equation in the form:
ax2 + bx + c = 0, we can factor using the
Quadratic Formula
 b  b 2  4ac
x
2a
USE Parenthesis
 (b)  (b) 2  4(a)(c)
x
2(a)
 (b)  (b) 2  4(a)(c)
x
2(a)
Notice the “plus/minus” notation.
This means you will have 2 answers for x when you are finished!!!!
 (b)  (b) 2  4(a)(c)
x
2(a)
 (b)  (b) 2  4(a)(c)
x
2(a)
Always make sure your equation is in standard form: ax2 + bx + c = 0
Example:
x2  4 x  21  0
Step 1:
Identify a, b, and c.
a=1
b=4
c = -21
Step 3:
Simplify Completely
 4  100
2
4  10
x
2
Step 2:
Plug in the values of a, b,
and c.
 (4)  (4)2  4(1)(21)
x
2(1)
Step 4:
You may need to show
answers as a Solution Set.
(just put curly q brackets
around the numbers with a
comma between them – in
order from least to
greatest.}
x
x
4  10
2
x3
x
4  10
2
x  7
7,3
Solve: 2x2 + x = 6
2x2  x  6  0
a2
b 1
c  6
 1  (1) 2  4(2)( 6)
x
2(2)
 1 49
x
4
1  7
4
 1  7 1  7 
x
,

4 
 4
x
6  8 3 
x   ,    ,2
4 4  2 
Notice, in the previous 2 examples the answers were real numbers.
Because of the square root step in the formula, we can have
different types of numbers as answers.
We can have 3 types of answers
1) Real numbers (2 different situations with real numbers)
2) Complex/Imaginary Numbers
Sometimes, we do not want to know what x is equal to, we just want to
know the TYPE of answers the equation has.
To find the type of answers an equation will have, we use the
Discriminant – please see next slide…
The discriminant is found using the part of the Quadratic Equation
that lives under the radical sign.
 (b)  (b) 2  4(a)(c)
x
2(a)
When you plug in a, b, and c – you will get an number. Use that
number to figure out what type of solutions the ORIGINAL
Quadratic Equation will produce., not the solutions themselves.
****The value you find from using the discriminant is just
helping us to find the types of solutions, it is not an answer
that you will use for anything else!!!!!!!!!!!!!!!!!!!!!!
(b)  4( a )(c )
2
b 2  4ac  0
b 2  4ac  0
If that answer is negative,
you have:
2 complex/imaginary roots
If that answer is exactly 0,
you have:
1 real root
b 2  4ac  0
If that answer is positive,
you have:
2 real roots
Solve x2 – 3x – 5 = 0
Always make sure your equation is in standard form: ax2 + bx + c = 0
Step 1:
Identify a, b, and c.
A=1
B = -3
C = -5
Step 2:
Plug in the values of a, b, and c.
Step 3:
Simplify Completely
Step 4:
You may need to show
answers as a Solution Set.
(just put curly q brackets
around the numbers with a
comma between them – in
order from least to
greatest.}
 (3)  (3) 2  4(1)(5) x  3  9  4(1)( 5)
x
2
2(1)
3  29
x
2
3  29
3  29
{x 
,x 
}
2
2
Note: There are 2 real, but
irrational roots! Why? Because
b2 - 4ac > 0
http://commons.wikimedia.org/wiki/File:Chalkandboard.jpg
Solve: x2 + 4x = -9
x2  4x  9  0
a 1
b4
c9
 4  (4) 2  4(1)(9)
x
2(1)
 4   20
 4  i 20
x
x
2
2
x
 4  2i 5
2
x  2  i 5
2 complex answers!
LINKS
Quadratic Formula
Explained
More Explanation
101 Uses for the Quadratic Equation