Geoscience Reference
In-Depth Information
5.3 DERIVATION OF THE QUADRATIC EQUATION FORMULA
The equation
ax
2
+
bx
+
c
= 0, where
a
,
b
, and
c
are any numbers, positive or negative, represents
any quadratic equation with one unknown. When this general equation is solved, the solution can be
used to determine the unknown value in any quadratic equation. The solution follows.
■
EXAMPLE 5.2
Problem:
Solve
ax
2
+
bx
+
c
= 0 for
x
.
Solution:
1. Subtract
c
from both members:
ax
2
+
bx
= -
c
2. Divide both members by
a
:
b
a
c
a
2
+
x
()
x
= −
3. Add (
b
/2
a
)
2
to both sides:
2
2
b
a
b
a
c
a
b
a
+
+
=− +
2
x
()
x
2
2
4. Complete the square:
2
2
b
a
c
a
b
a
=− +
x
+
2
2
5. Take the square root of both members:
2
b
a
c
a
b
a
+=±− +
x
2
2
6. Subtract
b
/2
a
from both members:
2
b
a
c
a
b
a
=− ±−+
x
2
2
Thus, the following quadratic formula is obtained:
2
=
−± −
bb ac
a
4
x
2
Search WWH ::
Custom Search