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:
ax2 + c = 0
ax2 = -c
You need both the positive and negative roots because x2 = x , 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)
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:
( ) 0
2 0
+ =
+ =
x ax b
ax bx
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 zeroproduct
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 x2 term is 1, then
x2 + bx + c = (x + n)(x + m), where n and m
i. Multiply to give c
ii. Add to give b
4. If the coefficient of the x2 term is not 1, then use either
a. Guess-and Check
i. List the factors of the coefficient of the x2 term
ii. List the factors of the constant term
iii. Test all the possible binomials you can make from these factors
b. Factoring by Grouping
i. Find the product ac
ii. Find two factors of ac that add to give b
iii. Split the middle term into the sum of two terms, using these two factors
iv. Group the terms into pairs
COMPLETING THE SQUARE
The technique of completing the square is presented here primarily to justify the
quadratic formula, which will be presented next. However, the technique does have
applications besides being used to derive the quadratic formula. In analytic geometry, for example, completing the square is used to put the equations of conic sections into standard form. Before considering the technique of completing the square, we must define a perfect square trinomial.
Perfect Square Trinomial
What happens when you square a binomial?
( )2 2 2 x + a = x + 2ax + a
· Note that the coefficient of the middle term (2a) is twice the square root of the
constant term (a2)
· Thus the constant term is the square of half the coefficient of x
· Important: These observations only hold true if the coefficient of x is 1.
This means that any trinomial that satisfies this condition is a perfect square. For
example,
x2 + 8x + 16
is a perfect square, because half the coefficient of x (which in this case is 4) happens to be
the square root of the constant term (16). That means that
x2 + 8x + 16 = (x + 4)2
Multiply out the binomial (x + 4) times itself and you will see that this works.
The technique of completing the square is to take a trinomial that is not a perfect square,
and make it into one by inserting the correct constant term (which is the square of half the
coefficient of x). Of course, inserting a new constant term has to be done in an
algebraically legal manner, which means that the same thing needs to be done to both
sides of the equation. This is best demonstrated with an example.
Example:
Given Equation: x2 + 6x - 2 = 0
Move original constant to other side: x2 + 6x = 2
Add new constant to both sides
(the square of half the coefficient of x): x2 + 6x + 9 = 2 + 9
Write left side as perfect square: (x + 3)2 = 11
Square root both sides
(remember to use plus-or-minus): x + 3 = ± 11
Solve for x: x = -3 ± 11
Notes
· Finds all real roots. Factoring can only find integer or rational roots.
· When you write it as a binomial squared, the constant in the binomial will be half of the coefficient of x.
THE QUADRATIC FORMULA
The solutions to a quadratic equation can be found directly from the quadratic formula.
The equation
ax2 + bx + c = 0
The advantage of using the formula is that it always works. The disadvantage is that it
can be more time-consuming than some of the methods previously discussed. As a
general rule you should look at a quadratic and see if it can be solved by taking square
roots; if not, then if it can be easily factored; and finally use the quadratic formula if there
is no easier way.
· Notice the plus-or-minus symbol (±) in the formula. This is how you get the
two different solutions—one using the plus sign, and one with the minus.
· Make sure the equation is written in standard form before reading off a, b, and
c.
· Most importantly, make sure the quadratic expression is equal to zero.
THE DISCRIMINANT
The formula requires you to take the square root of the expression b2 – 4ac, which is called the discriminant because it determines the nature of the solutions. For example, you can’t take the square root of a negative number, so if the discriminant is negative then there are no solutions.
