What
is a quadratic equation?
A quadratic equation is
an equation that can be
written in this form.
ax2+bx+c=0
The a,b, and c here represent
real number coefficients.
So this means we are talking
about an equation that
is a constant times the
variable squared plus
a constant times the variable
plus a constant equals
zero, where the coefficient
a on the variable squared
can't be zero, because
if it were then it would
be a linear equation.
Examples
2x2+3x+1=0,
x2+x=2x+3,
(x+2)(x+3)=5
All these equations are
equivalent to equations
of the above form. The
first one is already in
that form. The second
one can be put into it
by subtracting 2x+3 from
both sides. The third
one can be put into it
by multiplying out and
then subtracting 5 from
both sides.
Standard
Form
The form
ax2+bx+c=0
is the standard form
for a quadratic equation,
and for future reference,
here the letter a will
always mean the coefficient
on the square of the variable,
and b will be the coefficient
on the variable, and c
will be the constant term.
To get a quadratic into
standard form you must
remove all parentheses
and combine all like terms
and add or subtract something
from both sides so that
the right side will be
zero. Once you have your
equation in standard form
you can identify a,b,
and c.
Example
This and many of the
other examples below are
from my MathHelp collection
of problem sets, Quadratic
Equations. For more practice
and worked out examples
for this or any other techniques
explained here, click on
the MathHelp link at the
bottom of the page.
Problem: Write the equation
in standard form and identify
a,b, and c.
Solution: Multiply the
left side out and then
subtract the 5 from both
sides.
Solving
Now lets talk about solving
these equations.
Quadratic equations are
harder to solve than linear
equations, because once
you have them in standard
form it is hard to simplify
them any further, and
in this form there are
still two occurrences
of the variable, so it's
hard to see what we can
do to get the variable
alone.
So we have to find some
clever tricks to get around
this problem.
Solving
by Factoring
One trick is to solve the
equation by factoring. This
trick works because of the
principle of zero products.
The principle of zero products
says
If
A and B are real numbers
and AB=0, then either
A=0 or B=0.
This is a very special
property that only zero
has. For other numbers
there are lots of ways
to multiply and get them,
but not for zero. For
zero, the only way to
multiply numbers and get
it, is if one of the numbers
is zero.
The principle of zero
products allows us to
reduce a complicated equation
to simpler equations provided
the right side of the
equation is zero, and
the equation is factored,
because we can set each
of the factors equal to
zero.
- To solve a quadratic
by factoring, first
you must make sure it
is in standard form.
It
is especially important
that it is set equal
to zero,
because remember, the
principle of zero products
only works for zero.
- Then you must factor
the left side.
- Then you set each
of your factors equal
to zero and solve the
equations you get to
find the solutions to
your equation.
Example
Problem: Solve the equation.
Solution:
The second line here comes
from setting x+1 and x+6
equal to 0 by the principle
of zero factors.
What
if you can't factor?
But some quadratics are
difficult to factor, so
for these equations we need
other methods. The method
of completing the square
is a method that will work
for any quadratic, but it
is a little bit complicated,
so I will introduce it slowly
and step by step. But first
to give you an overview
of where we are going, I
will show you a simple example
of it.
Consider the following
equation.
x2-2x-1=0
This looks like a nice
simple friendly equation,
but we can't solve it
by factoring, because
we can't find two numbers
to multiply and get -1
and add and get -2, so
we are going to have to
find another method.
But if only that minus
sign on the 1 weren't
there, then we could factor
it really easily, in fact
it would be a perfect
square. How can we make
that minus sign go away?
Well, one thing we could
do is add 2 to both sides
of the equation, and then
the equation would become
x2-2x+1=2
and this factors to
(x-1)2=2
Now, I know what you're
saying. You are saying,
"But you said that you
have to get it set equal
to 0 to solve it by factoring,
because the principle
of zero products only
works for 0. What good
does it do to have something
factored and set equal
to 2?"
And if you are saying
this to yourself, you
are absolutely right.
But this isn't just any
old factorization. It
is a perfect
square, and
maybe you can do something
with a perfect square
set equal to 2.
If we could figure out
how to solve equations
like
(x-1)2=2
that is, perfect squares
set equal to numbers,
then we could solve an
equation like
x2-2x-1=0.
And if we could find
a way to add a number
to both sides of other
quadratics so that we
can put them into the
form perfect square equals
constant, then maybe we
could be able to solve
them too.
This means that to help
us solve quadratic equations,
we need to learn two skills.
- Solve equations
of the form
(x+k)2=d,
where k and
d are numbers.
- Find a way
of figuring
out what number
to add to
both sides
of a quadratic
equations
so that the
left side
will become
a perfect
square.
|
To work our way up to
the task of solving equations
of the form
(x+k)2=d
let's first start with
the slightly easier task
of solving equation of
the form
x2=d
How do we solve an equation
of the form
x2=d?
If x is greater than
0 then the obvious answer
is
but this is not quite
right because it only
gives you the positive
square root of d, and
all positive numbers have
two square roots, a positive
one and a negative one.
So to be sure that you
are getting all solutions
to an equation of this
form, your answer must
be
If x less than 0 then
what happens? What kind
of number can you square
and get a negative number?
If you square a positive
number then clearly you
get a positive number.
But if you square a negative
number then you have a
product of two negative
numbers, so you still
get a positive number.
So what is left for squaring
and getting a negative
number? Nothing. So the
equation has no solutions.
Now let's look at the
more general equation
of the form
(x+k)2=d
This is really not much
harder since anything
you can do with x you
should be able to do with
x+k. x+k represents a
number too. So solve for
x+k and then add something
to both sides of the equation
to get x alone.
Example 1
Problem: Solve the equation.
Solution:
Example 2
Problem: Solve the equation.
Solution: Anything you can
do with x, you can do with
x+1, and once you find x+1
all you have to do to get
x is subtract 1.
Completing
the Square
Now to problem number two,
that of finding something
to add to a quadratic to
make it a perfect square.
This is what is meant
by completing the square,
and the secret to it is
to expand out the expression
(x+k)2
and see what makes perfect
squares tick. Applying
our formula for squaring
a binomial, we get
(x+k)2=x2+2kx+k2
The key here is to look
at the relationship between
the coefficient on x and
the constant coefficient.
The coefficient on x is
2k and the constant term
is k2. This
means that if we know
the coefficient on x,
and we want to know what
the constant term has
to be for the expression
to be a perfect square,
then we need to divide
the coefficient on x by
2 to get k, and then square
to get k2.
So if you have an expression
of the form
x2+bx
and you want to find
something to add to it
to make it a perfect square,
then you need to
- Divide b
by 2 to get
k
- Square k
to get k2.
|
Example
Problem: Complete the square.
Solution:
The yellow part is the scratch
paper. On the scratch paper
you first divide the 9 by
2 and then square the result.
Don't worry about the minus
sign, because it will go
away when you square anyway.
Then the number you get
will be the number you need
to add to the expression
to make a perfect square
out of it. After you do
that it is good practice
to write it as the square
that it is. For that you
can use the first line of
your scratch paper and match
the sign with the sign of
the second term of the original
expression.
I hope the above has
helped you understand
the process of completing
the square. If not, there
is another approach to
it that I have written
an article about that
you might find interesting
for further understanding.
It is a geometrical approach
based on the method that
many earlier mathematician
used. You can read my
article A Geometrical
Approach to Completing
the Square to find
out about it.
Solving
by Completing the Square
Now we are ready to use
the method of completing
the squares to solve quadratic
equations. The best way
to do this is as follows.
- Add something
to both sides
so that the
left side
has no constant
term.
- Figure out
what to add
to the left
side to make
it a perfect
square, and
add that to
both sides.
- Write the
left side
as the perfect
square that
it is and
do the arithmetic
on the right
side.
- Solve the
equation you
get by the
methods of
equations
of the form
(x+k)2=d
|
One thing we left out.
So far all of the equations
we have solved have had
a coefficient of 1 on
x2. What do
we do if we have a coefficient
other than 1 on there?
Well, we don't really
have any method of completing
the squares to deal with
that situation, so the
easiest thing to do is
just divide both sides
by it and put up with
the fractions. With completing
the squares, fractions
are not so bad to deal
with because there is
no guess work.
Example
Problem: Solve the equation.
Solution:
First we divide both sides
of the equation by 2, to
get a coefficient of 1 on
the first term. 0/2 is still
0. Then we add 1/2 to both
sides so that it is easier
to figure out what to add
to make the left side a
perfect square. Then we
complete the square in order
to figure out what to add
to both sides to make the
left side a perfect square.
The scratch work for this
is shown in yellow. The
best way to divide a fraction
by 2 is to multiply it by
1/2. Then we write the left
side as the perfect square
that it is, and do the arithmetic
on the right side. Now take
square roots and add 3/4
to both sides to get the
final answer.
The
Quadratic Formula
Now that you have learned
the method of completing
the squares, I will tell
you a secret. The methods
of completing the squares
is such a good method for
solving quadratics that
it is very seldom used for
it. It is used for other
things in mathematics. See
my article
Equations
of Circles for
another use for it. But
for solving quadratic equations,
it is such a good method
that it puts itself out
of business.
You see, with such a
mechanical method like
the method of completing
the squares, why not just
apply it to the general
quadratic equation and
solve all quadratics in
the world at once, and
be done with it, and never
have to use algebra to
solve a quadratic again.
Problem: Solve the equation:
Solution:
Just do it the same way
with the letters as you
did with the numbers.
First divide both sides
by a. Then subtract c/a
from both sides to be
able to see more easily
what to add to the left
side to make it a perfect
square. Then complete
the square on the scratch
paper and add what you
get to both sides. Write
the left side as the perfect
square that it is. Instead
of doing arithmetic on
the right side you have
to do a little bit of
algebra, using 4a2
as a common denominator.
Then take square roots
and subtract b/2a from
both sides and use 2a
a common denominator to
get the final answer.
We have just solved all
quadratics in the world
at once and derived the
quadratic formula, which
says:
For any real numbers
a,b, and c, the solutions
to the equation
ax2+bx+c=0
are
Solving
by the Quadratic Formula
Since this formula is
somewhat long and complicated,
it is best to evaluate
it in two smaller pieces
by first evaluating the
thing inside the radical,
b2-4ac
and then put the result
into the formula
The quantity
b2-4ac
even has a name. It is
called the discriminant.
And there is another advantage
to computing it first.
Since it is what is in
the radical, it can't
be negative if there are
going to be solutions
to the equation, because
you can't take a square
root of a negative number,
(unless you use the imaginary
numbers, and we're not
yet ready for them here)
so if the discriminant
comes out negative, then
you don't have to do any
more work, and all you
have to do is write "no
solution" on your paper
and you are done. Sometimes
you can determine this
quite quickly by estimating,
particularly if a and
c are very large and b
is small.
Example 1
Problem: Solve the equation.
Solution:
Example 2
Problem: Solve the equation.
Solution: No solution. b
is -2, which is very small
in comparison to a=6 and
c=27, so you don't even
have to compute the discriminant
to see that it is going
to be negative.
Imaginary Solutions
This section is for more
advanced students who know
about imaginary numbers.
For a brief introduction
see my article
Complex Number
Arithmetic. If you know
about imaginary numbers,
you don't have to stop when
you see the square root
of a negative number, because
with imaginary numbers you
can take the square root
of a negative number. To
find the square root of
any negative number you
just take the square root
of the corresponding positive
number and multiply it by
i, the square root of -1.
This makes sense at least
once you believe in the
idea that the square root
of -1 is i, because of the
multiplication rule for
square roots. (See
Square
Roots.)
(It is customary usually
to write the real number
after the i when it is a
square root so that it is
clear that the i is not
inside the radical.)
Once you know how to
find square roots of negative
numbers, you find imaginary
solutions to quadratics
by the completing the
square or the quadratic
formula pretty much like
you find real ones. For
the following examples
the instruction is to
solve the equation.
Example 1:
x2+2x+5=0
Solution:
Completing
the Square:
Explanation:
First we add -5 to both
sides to get the constant
on the right side of the
equation so that it is more
clear what we need to add
to the left side to make
it a perfect square. Then
we complete the square.
2/2=1, 1
2=1,
so 1 is the number we add
to the left side to get
a perfect square. Whatever
you do to one side of an
equation, you have to do
to the other side, so we
also add 1 to the right
side of the equation. Then
we write the left side as
the perfect square that
it is, and do the arithmetic
on the right side. In this
equation we get a negative
number on the right side,
but with imaginary numbers
we can deal with that. The
two square roots of -4 are
2i and -2i, so x+1 has to
be one of them. Then to
find out what x is, all
we have to do is add -1
to both sides.
Quadratic
Formula:
Explanation:
First compute the discriminant
and find its square root.
The square root of -16 is
the square root of 16 times
i, 4i. Then we just fit
in the square root in its
place in the formula. The
numerator has a common factor
of 2 that we can factor
out and cancel with the
2 in the denominator.
Example 2:
x2+x+1=0
Completing
the Square:
Explanation:
Again we first subtract
1 from both sides so that
it is more clear what we
need to add. For the completing
the square part, this time
the coefficient on x is
1. Half of 1 is 1/2 and
(1/2)
2=1/4, so
1/4 is the completing the
square number. We add it
to both sides, because anything
you add to one side you
have to add to the other
side. Then again we get
a negative number to take
plus and minus square roots
of, but we can handle that
with imaginary numbers.
We just break it up into
square roots of -1, 3, and
4. The square root of -1
is i, the square root of
4 is 2, and the square root
of 3 is the square root
of 3. Then to get the final
answer we add -1/2 to both
sides.
Quadratic
Formula:
Explanation:
First we figure out the
discriminant, which comes
out to be -3, and then we
take its square root. The
square root of a negative
number is just the square
root of the corresponding
positive one times i. This
time we can't the square
root of 3 is an irrational
number, so it is better
left undone thinking of
radical 3 is the name for
the exact number that you
multiply by itself to get
3. Then after we have found
the square root of -3, we
can put it in place of the
radical in the formula to
get our final answer.
Example 3:
3x2+2x+3=0
Completing
the Square:
Explanation:
In this one there is a coefficient
other than 1 on the x
2,
namely 3, so first we have
to get rid of it by dividing
both sides by 3. Then from
there it is pretty much
like the other examples.
Next we add -1 to both sides
to see more clearly what
we need to add to make the
left side a perfect square.
The coefficient on x is
2/3. To find half of 2/3
we multiply it by 1/2 and
get 1/3. Then to find the
completing the square number,
we square that and get 1/9,
which we add to both sides
of the equation. Then we
write the left side as the
perfect square that it is
and do the arithmetic on
the right side. And again
since this section is about
imaginary solutions, we
get a negative number on
the right side of the equation
to take the square root
of. Then after taking plus
and minus square roots,
we add -1/3 to both sides
to get the final answer.
Quadratic
Formula:
Explanation:
First we find the discriminant,
and we get -32. The square
root of -32 is the square
root of -1 times the square
root of 16 times the square
root of 2. The square root
of -1 is i, the square root
of 16 is 4, and the square
root of 2 is something nasty
and irrational, so it is
left as the square root
of 2. Then we put this square
root in place of the radical
in the formula and simplify
to get the final answer.
