Solving Quadratic Equations
An equation that can be simplified to the form
is called a quadratic equation. If the
equation is already simplified to this form we say it is in standard form. We
have already investigated two techniques for solving quadratic equations:
extraction of roots (which works if there is no linear term) and factoring.
Neither technique is practical in all cases. We will now investigate a technique
that is practical more generally.
Using the Quadratic Formula
The quadratic formula lets us find the solutions of a quadratic equation
when it is written in standard
form. The plus or minus symbol (±) allows us
to represent both solutions in a single expression. Lets take a look at an
example to see how it works.
Example: Use the quadratic formula to find the solutions
to the equation
Solution: First we identify the coefficients a, b and c
for this equation: a = 3, b = 4 and c = −5. Then we substitute these values for
a, b and c in the formula:
Let’s represent each solution separately and use our
calculators to approximate the solutions.
We can enter each of these solutions into our graphing
calculators with a single entry but we must pay careful attention to the order
of operations. Let’s start with the first solution. Enter it into your
calculator as follows:
When you push enter the calculator should return the value
.7862996478
This is an approximation of the first solution. Now we
turn our attention to the second solution. Press 2^{nd} Enter to bring up the
previous entry:
The entry for the second solution is identical except that
the addition that I’ve highlighted above should be changed to a subtraction:
When you push enter the calculator should return the value
−2.119632981
So the solutions to the equation are approximately x =
.7862996478 and x = −2.119632981
