Derivation Of The Quadratic Formula Worksheet With Answer Key Page 5
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5Algebra 1
Name___________________
Derivation of the Quadratic Formula
After today’s lesson, you should know the quadratic formula and be familiar with its proof by completing the
square. You should also be able to solve quadratic equations by using the quadratic formula. (CA 19.0, 20.0)
Algebraic Representations
Directions
Divide the general form of the quadratic
b
c
≠
= 0 ,
a
0
2
x
x +
+
equation by a.
Step 1:
a
a
b
c
c
+
=
−
2
Subtracted the constant
from both sides of
x
x
Step 2:
a
a
a
€
the equation.
Take half of the coefficient of the linear term,
2
2
⎛
⎞
⎛
⎞
b
b
c
b
+
+
=
−
+
2
x
x
⎜
⎟
⎜
⎟
squared it, and added it to both sides of the
Step 3:
€
a
⎝
2
a
⎠
a
⎝
2
a
⎠
equation.
Factor the trinomial on the left side of the
2
2
⎛
⎞
⎛
⎞
b
c
b
+
=
−
+
⎜
⎟
⎜
⎟
x
equation.
Step 4:
⎝
2
a
⎠
a
⎝
2
a
⎠
2
2
⎛
⎞
2
b
b
c
b
"
$
+
=
−
+
Multiply out
on the right side of the
⎜
⎟
x
Step 5:
#
%
2
2a
⎝
2
a
⎠
a
4
a
equation.
c
2
2
⎛
⎞
b
c
4
a
b
Multiply
by 1 to obtain common
−
+
=
−
•
+
⎜
⎟
x
Step 6:
a
2
⎝
2
a
⎠
a
4
a
4
a
€
denominators.
Combine the fractions in the right side of the
2
−
2
⎛
b
⎞
b
4
ac
+
=
⎜
⎟
x
equation.
Step 7:
€
2
⎝
2
a
⎠
4
a
Take the square root of both sides of the
2
−
2
⎛
b
⎞
b
4
ac
+
=
±
equation.
⎜
x
⎟
Step 8:
2
⎝
2
a
⎠
4
a
−
2
2
⎛
⎞
b
b
4
ac
b
"
$
+
=
±
⎜
x
⎟
Simplify
on the left side of the
x +
Step 9:
2
⎝
2
a
⎠
4
a
#
%
2a
equation.
−
2
a
a
b
b
4
ac
+
=
±
Use the property
on the right side
x
=
Step 10:
b
b
2
a
2
4
a
€
of the equation.
−
2
2
b
b
4
ac
Simplify
on the right side of the
4a
+
=
±
x
Step 11:
equation.
2
a
2
a
€
b
−
2
b
b
4
ac
Subtract
from both sides of the equation.
€
=
−
±
x
Step 12:
2a
2
a
2
a
Combine the fractions to obtain the Quadratic
−
±
−
2
b
b
4
ac
=
x
Formula.
Step 13:
€
2
a
MCC@WCCUSD 12/02/11
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