Definition
ax2 + bx + c = 0
a, b, c are constants (generally integers)
Roots
Synonyms: Solutions or Zeros
Can have 0, 1, or 2 real roots
Consider the graph of quadratic equations. The
quadratic equation looks like ax2 + bx + c = 0, but if
we take the quadratic ex pression on the left and set
it equal to y, we will have a function:
y = ax2 + bx + c
When we graph y vs. x, we find that we get a curve
called a parabola. The specific values of a, b, and c
control where the curve is relative to the origin (left,
right, up, or down), and how rapidly it spreads out.
Also, if a is negative then the parabola will be
upside-down. What does this have to do with
finding the solutions to our original quadratic
equation? Well, whenever y = 0 then the equation
y = ax2 + bx + c is the same as our original equation.
Graphically, y is zero whenever the curve crosses
the x-axis. Thus, the solutions to the original
quadratic equation (ax2 + bx + c = 0) are the values
of x where the function (y = ax2 + bx + c) crosses
the x-axis. From the figures below, you can see that
it can cross the x-axis once, twice, or not at all.
Actually, if you have a graphing calculator this
technique can be used to find solutions to any
equation, not just quadratics. All you need to do is
1. Move all the terms to one side , so that it is equal
to zero
2. Set the resulting expression equal to y (in place of
zero)
3. Enter the function into your calculator and graph
it
4. Look for places where the graph crosses the x-axis
Your graphing calculator most likely has a function
that will automatically find these intercepts and give
you the x-values with great precision. Of course, no
matter how many decimal places you have it is still
just an approximation of the exact solution. In real
life, though, a close approximation is often good
enough.
Solving Quadratic Equations
A. Solving by Square Roots
No First-Degree Term
If the quadratic has no linear, or first-degree term
(i.e. b = 0), then it can be solved by isolating the x2
and taking square roots of both sides:
∞ You need both the positive and negative roots
because 
, so x could be
either positive or
negative.
∞ This is only going to give a real solution if
either a or c is negative (but not both)
B. Solving by Factoring
Solving a quadratic (or any kind of equation) by
factoring it makes use of a principle known as the
zero-product rule.
Zero Product Rule
If ab = 0 then either a = 0 or b = 0 (or both).
In other words, if the product of two things is zero
then one of those two things must be zero, because
the only way to multiply something and get zero is
to multiply it by zero.
Thus, if you can factor an expression that is equal to
zero, then you can set each factor equal to zero and
solve it for the unknown .
The expression must be set equal to zero to use
this principle
You can always make any equation equal to
zero by moving all the terms to one side.
Example:
Given:
x2 – x = 6
Move all terms to one side:
x2 – x – 6 = 0
Factor:
(x – 3)(x + 2) = 0
Set each factor equal to zero and solve:
(x – 3) = 0 OR (x + 2) = 0
Solutions:
x = 3 OR x = -2
No Constant Term
If a quadratic equation has no constant term (i.e.
c = 0) then it can easily be solved by factoring out
the common x from the remaining two terms:
Then, using the zero-product rule, you set each
factor equal to zero and solve to get the two
solutions:
x = 0 or ax + b = 0
x = 0 or x = –b/a
WARNING: Do not divide out the common factor
of x or you will lose the x = 0 solution. Keep all the
factors and use the zero-product rule to get the
solutions.
Trinomials
When a quadratic has all three terms, you can still
solve it with the zero-product rule if you are able to
factor the trinomial.
Remember, not all trinomial quadratics can be
factored with integer constants
If it can be factored, then it can be written as a
product of two binomials. The zero-product rule can
then be used to set each of these factors equal to
zero, resulting in two equations that are both simple
linear equations that can be solved for x. See the
above example for the zero-product rule to see how
this works.
A more thorough discussion of factoring trinomials
may be found in the chapter on polynomials , but
here is a quick review:
Tips for Factoring Trinomials
1) Clear fractions (by multiplying through by the
common denominator)
2) Remove common factors if possible
3) If the coefficient of the
term is 1, then
4) x2 + bx + c = (x + n)(x + m), where n and m
i. Multiply to give c
ii. Add to give b
5) If the coefficient of the
term is not 1, then
use either
i. Guess-and Check
ii. List the factors of the coefficient of the
term
iii. List the factors of the constant term
iv. Test all the possible binomials you can
make from these factors
v. Factoring by Grouping
6) Find the product ac
i. Find two factors of ac that add to give b
ii. Split the middle term into the sum of two
terms , using these two factors
iii. Group the terms into pairs
iv. Factor out the common binomial
