Geoscience Reference
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5.2 QUADRATIC EQUATIONS: THEORY AND APPLICATION
The equation 6 x = 12 is a form of equation familiar to most of us. In this equation the unknown
appears only in the first degree, so it is a simple equation or linear equation. Those experienced in
mathematics know that not all equations reduce to this form. For instance, when an equation has been
reduced, the result may be an equation in which the square of the unknown equals some number, as
in x 2 = 5. In this equation, the unknown appears only in the second degree, so it is a pure quadratic
equation. In some cases, when an equation is simplified and reduced, the resulting equation contains
the square and first power of the unknown, which equal some number, such as x 2 - 5 x = 24. An equa-
tion containing the first and second degree of an unknown is an affected quadratic equation.
Quadratic equations, and certain other forms, can be solved with the aid of factoring. The proce-
dure for solving a quadratic equation by factoring is as follows:
1. Collect all terms on the left and simplify to the form ax 2 + bx + c = 0.
2. Factor the quadratic expression.
3. Set each factor equal to zero.
4. Solve the resulting linear equations.
5. Check the solution in the original equation.
EXAMPLE 5.1
Problem: Solve x 2 - x - 12 = 0.
Solution:
1. Factor the quadratic expression.
(
x
−+=
4
)(
3
)
0
2. Set each factor equal to zero.
x
−= +=
40
x
30
3. Solve the resulting linear equations.
x
=
4
x
= −
3
Thus, the roots are x = 4 and x = -3.
4. Check the solution in the original equation.
2
2
()
4412 0
00
−− =
=
() ()
−−−− =
=
3
3
12 0
00
Many times factoring is either too time consuming or not possible. The formula shown below is
called the quadratic formula. It expresses the quadratic equation in terms of its coefficients. The
quadratic formula allows us to quickly solve for x with no factoring.
2
= −± −
bb ac
a
4
x
(5.1)
2
To use the quadratic equation, just substitute the appropriate coefficients into the equation and solve.
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