Math-ML: Deriving the Quadratic Formula
This page outlines the derivation of the Quadratic Formula.
We take a quadratic equation in its general form, and solve for x:
a
x
2
+
b
x
+
c
=
0
a
x
2
+
b
x
=
-
c
x
2
+
b
a
x
=
-
c
a
Divide out leading coefficient.
x
2
+
b
a
x
+
b
2
a
2
=
-
c
(
4
a
)
a
(
4
a
)
+
b
2
4
a
2
Complete the square.
(
x
+
b
2
a
)
(
x
+
b
2
a
)
=
b
2
-
4
a
c
4
a
2
Discriminant revealed.
(
x
+
b
2
a
)
2
=
b
2
-
4
a
c
4
a
2
x
+
b
2
a
=
b
2
-
4
a
c
4
a
2
x
=
-
b
2
a
±
{
C
}
b
2
-
4
a
c
4
a
2
There's the vertex formula.
x
=
-
b
±
{
C
}
b
2
-
4
a
c
2
a