Try our Free Online Math Solver!

Solving Equations IExpressions Involving Power Functions
Applying the Toolbox— Solving Equations
Example 2.5 Solve the equation 3x(x +1) = (x +1) ( x^{2 } + 2).
Guess and plug in method. The following table gives the outcome of plugging
in
various numbers into the left hand and righthand sides of this equation.
Here the value of the left hand side
equals the righthand side.
So far, we’ve identified that when x = 1, the lefthand
side of the equation equals the righthand
side. Are there any other values of x which do the same? How do you know? This
is
a very poor method.
Graphing method to estimate solutions. Instead of guessing which numbers to
plug in, we
could graph y = 3x(x +1) and y = (x + 1) ( x^{2} + 2) on the same set
of axes to get a more
complete picture. Finding the values of x which solve the equation is the same
as finding
the xcoordinates of the points where the graphs of y = 3x(x +1) and y = (x +
1) ( x^{2}+ 2)
intersect (or cross).
Use your graphing calculator to graph y = 3x(x +1) and y = (x + 1) ( x^{2}+
2) on the same
axes. Change your viewing rectangle to make sure you are seeing all the
important features
of your graphs. Trace and zoom to estimate the xcoordinates of the points where
the two
graphs cross. Are any of your answers close to x = 1? Are there any other
solutions?
Questions to ponder over the semester: How do you know that you have found all
the
solutions? Is your viewing rectangle large enough? These questions are not so
easy to
answer, but you will be able to by the end of the semester.
Using a graphing calculator will only allow you to estimate the solutions. We
need to use
algebra to find exact solutions. But using both graphing and algebra in the
exercises will
increase your understanding and allow you to check your answers.
Algebraic method. We’ll use our toolbox in a series of steps.
Preliminary work. Get all terms which are on the righthand side of the equation
over to
the left by adding or subtracting them to each side. With all your mental
powers, resist
the temptation to cancel out anything by dividing. You will only miss possible
solutions.
Here we add (x +1)( x^{2}+ 2) to each side to get
3x(x +1) + (x + 1)(x^{2} + 2) = 0.
Combine all like terms. Deconstruct the lefthand side—play with it! How do you
know
when you have what you need? Experience and practice! Our end goal is to turn
the left
side into a product of expressions and then use the Zero Law.
One way to deconstruct the left part of the lefthand side is to see it as a
product of
x +1and 3x . The right part is the product of x +1 and x^{2} + 2. We
have identified a
common factor of x +1:
identify the common term,  
factor it out. 
Now at this stage we pause and check: are there any other
common terms in the leftover
pieces? There are no other common terms but if there were, we would factor them
out in
the same way. Continue the process of identifying common terms and factoring
them
out, until there are no more. Then add or subtract the leftovers. Our equation
now looks
like
(x +1)( x^{2} 3x + 2) = 0.
This quadratic term can be factored (always check for this), so we end up with
(x +1)(x  2)(x 1) = 0.
Use the Zero Law. We did this part of the problem in Example 2.4. The Zero Law
says
that the solutions of (x +1)(x  2)(x 1) = 0 are the same as the solutions of x
+1 = 0,
x  2 = 0, and x 1 = 0. Thus our final answers are x = 1, 1, 2.
How do these solutions compare to the answers you estimated with your graphing
calculator?
Example 2.6 Solve the equation
Graphing Method. Following the previous example we
see that we want to graph the
lefthand side and the righthand side on the same set of axes. We can estimate
the
solutions to this equation by tracing and zooming in on the points of
intersection. Try
this in this case. Remember that y = 0 is just the xaxis. What happens near x =
4 ? In
fact, your calculator is confused—it has no idea what to do at x = 4 .
Using our brains, we see that plugging in x = 4 gives a zero denominator. This
means
that the lefthand side of the equation is undefined at x = 4 .
We can graph the lefthand side with our graphing calculators as long as we have
the
calculator avoid x = 4 . To do this, we must graph the function in two
pieces —first with
x values smaller than 4, and then with x values larger than 4.
Step 1: Graph y = 0 and
on the same set of axes using a viewing window with a
maximum xcoordinate of 3.5 or
3.99 (or anything just a bit smaller than 4).
Step 2: Graph these same functions using a window with a minimum xcoordinate of
4.01 or 4.1 (or something just a bit larger than 4).
In each step trace and zoom to estimate the xcoordinates of any place where
crosses the xaxis (which happens to be the second
function y = 0 ).
Algebraic Method. Here we already have everything but zero on the
lefthand side.
Since the lefthand side contains algebraic fractions, we want to multiply
through (both
sides of the equation) by the common denominator to get rid of the algebraic
fractions.
In this case the common denominator is (x  4)^{4} . We now want to
solve this revised
equation.
Any xvalue which is a solution of the original equation will be a solution to
this revised
equation. This means that we will find the solutions to the original equation
among those
to the revised equation
Read that last sentence carefully—a very important fact is
that in solving the revised
equation, we may get xvalues which are not solutions to the original equation.
So we
have a twostep process: solve the revised equation, then check to see which of
these
values are really solutions to the original equation. This last step is called
checking for
extraneous solutions.
Solve revised equation. Solve the equation
This is one complicated algebraic expression subtracted
from another. When your
equation is of this form, combining like terms will often help simplify it . We
will follow
the form of the examples in the Toolbox.
First notice that there is a factor of both x  4 and x + 2 in each side of the
minus
symbol. We will take the smallest power of each of these terms as our common
term.
The smallest power of x  4 is 1 and the smallest power of x + 2 is
. With the
common terms underlined, our equation looks like
We now factor out the common terms and subtract the
leftovers in the following series of
steps.
At this point the lefthand side of the equation is now
the product of three simpler
equations. Use the Zero Law! The solutions to this equation are the solutions to
the three
equations
The first and third are linear equations yielding the
solutions x = 4 and x = 28
respectively. The middle equation can be solved by raising each side to the
power
,
which is the reciprocal of the power
. Thus its solution
is x = 2
Remember our twostep process! These xvalues may be the solutions to the
revised
equation, but they may not even be in the domain of the lefthand side function.
We have
to worry about zeros in the denominator as well as negative numbers under a
square root ,
fourth root, etc.
Checking for extraneous solutions. The easiest way to do this is to just start
plugging in
to the original equation the (possibly false) solutions that you have already
obtained.
When you plug in an xvalue, do you really get zero on the left? Or must you
conclude
the lefthand side is undefined at that xvalue? We see that if we plug in x =
2 and
x = 28 to the original equation we really do get 0 = 0 . But we have already
seen that
x = 4 gives a zero denominator. So our solutions to the original equation are
only
x = 2,28. How do these check out with the solutions you estimated with the
graphing
method?
Summary : Solving Equations 
Preliminary work. Get all terms to the lefthand
side of the equation, leaving only zero
on the righthand side. Convert all negative exponents into positive exponents.
If you
have algebraic fractions, multiply through by the common denominator, and
continue to
the next step to solve this revised equation.
Tips on solving equations without algebraic fractions. It generally helps
to keep
asking yourself the following series of questions in order over and over again.
If you
answer yes to any of them, follow the instructions.
Q1. Basic Questions: Is the lefthand side of the equation
a linear, quadratic, or other
polynomial that can be factored ? If yes, …you know what to do. Is the entire
lefthand
side raised to a power? If yes, raise each side to the reciprocal of that power
to
get rid of it, and then repeat question Q1.
Q2. Is the lefthand side of the equation a product of algebraic expressions? If
yes, use
the Zero Law to break the original equation into smaller equations. Repeat the
question process (starting with Q1) to solve each smaller equation.
Q3. Are there any like terms? If yes, combine them following the examples in the
toolbox. Go back to Q2.
If these steps do not lead to a complete solution, you should try anything else
you can
think of. But you might have to rely on the numerical estimates you obtained
from
graphing (or some other source).
Check for extraneous solutions. Plug in to the original equation all
solutions obtained
above. Do you get zero in the denominator? Or a negative number under a square
root,
fourth root, etc? Later on in the course, we will see other ways extraneous
solutions will
occur. Any solution which gives any kind of problem like this is NOT really a
solution.
All other solutions are your answers. Check them against the graphing estimates.
Prev  Next 