Graphics Reference
In-Depth Information
3.3 Solving the Roots of a Quadratic Equation
To put the above laws and notation into practice, let's take a simple example
to illustrate the logical steps in solving a problem. The task involves solving
the roots of a quadratic equation, i.e. those values of
x
that make the equation
equal zero.
Given the quadratic equation where
a
=0:
ax
2
+
bx
+
c
=0
Step 1
: subtract
c
from both sides:
ax
2
+
bx
=
−
c
Step 2
: divide both sides by
a
:
x
2
+
b
c
a
a
x
=
−
b
2
4
a
2
Step 3
:add
to both sides:
b
2
4
a
2
b
2
4
a
2
x
2
+
b
c
a
+
a
x
+
=
−
Step 4
: factorize the left side:
x
+
2
b
2
a
c
a
+
b
2
4
a
2
=
−
Step 5
:make4
a
2
the common denominator for the right side:
x
+
2
4
ac
+
b
2
4
a
2
b
2
a
=
−
Step 6
: take the square root of both sides:
√
b
2
2
a
=
±
b
−
4
ac
x
+
2
a
b
2
a
Step 7
: subtract
from both sides:
x
=
±
√
b
2
−
4
ac
2
a
b
2
a
−
Step 8
: rearrange the right side:
√
b
2
x
=
−
b
±
−
4
ac
(3.9)
2
a
This last expression gives the roots for any quadratic equation.
