deriving the quadratic formula
DESCRIPTION
Deriving the quadratic formula from the quadratic equationTRANSCRIPT
© D. T. Simmons, 2009 1
From Whence
The Quadratic Formula?
Don Simmons
© D. T. Simmons, 2009 2
Deriving the Quadratic Formula
Everybody taking algebra eventually learns the quadratic
formula. But few know where it comes from.
This presentation will help you understand how to get …
2
2
From: 0
4To:
2
ax bx c
b b acx
a
© D. T. Simmons, 2009 3
Deriving the Quadratic Formula
• Begin with the quadratic equation…
• Set the leading coefficient to one.
2 0 ax bx c
2
2
0
0
a
ax bx c
x xb c
a a
© D. T. Simmons, 2009 4
Deriving the Quadratic Formula
• Isolate the variables on one side of the equation by
moving the c value.
2
2
0b c
x xa a
bx x
a
c
a
© D. T. Simmons, 2009 5
Deriving the Quadratic Formula
• Complete the square.
2
2
2
2
2 2
b cx x
a a
b cx x
a
b
a
b
a a
© D. T. Simmons, 2009 6
Deriving the Quadratic Formula
• Create a common denominator.
• Combine the fractions
2 22
2 24 4
4
4
cb b bx x
a
a
a a a a
22
2
2
2
4
4 4
b b abx
a
cx
aa
© D. T. Simmons, 2009 7
Deriving the Quadratic Formula
• Rewrite in exponential form.
222
2 2
2
2
2
4
4 4
4
42
bx
b a cb bx x
a a
a
a
b a c
a
© D. T. Simmons, 2009 8
Deriving the Quadratic Formula
• Apply the square root property
2 2
2
2
4
2 4
4
2 2
b a cbx
a a
b a cbx
a a
© D. T. Simmons, 2009 9
Deriving the Quadratic Formula
• Isolate x.
2
2
4
22
2
22
4
b b
a
b a cbx
a a
b
a
b a cx
a
© D. T. Simmons, 2009 10
Deriving the Quadratic Formula
• Done!
2 4
2
b b a cx
a