2. Solve for x and check.
First clear denominators. In this case that amounts to cross
Now that we have gotten rid of denominators, we will want to
simplify both sides. Remove parentheses.
At this point it looks like we have a quadratic equation to
solve. The technique for solving a quadratic equation is to transpose
all terms to one side leaving a 0 on the other, and then factoring, if
possible, of using the quadratic formula if not. However, when we
subtract x squared from both sides,
we find that our work has simplified down to a first degree equation.
In this case we transpose known terms to one side, and unknown
terms to the other side. Terms that have an x in them are unknown
terms and terms that don't have an x in them are known terms.
At this point, all of the terms on the left side have a factor of x, so
we can factor it out.
This technique gets all of our x's together in one place with only one
thing happening to it. To get rid of the coefficient that is multiplying
our x, divide both sides by the coefficient of the unknown.
This is a formula which will give us a solution for any equation
which is of the same form as the equation which we are solving. Note
that such an equation will not have a solution if the denominator in
this solution, d + a - c - b = 0. That is to say, if a - b = c - d, then
the equation will not have a solution. In that case, the tops and
bottoms of the two sides of the equation will not only have the same
ratios, they will also have the same differences. The only way that
that can happen is if a = c, and b = d. In that case, any value of x
except for the common value of b or d will be a solution, and the
solution will not be unique. Otherwise, there will be no solution.
There are other possible problems which will not be evident until we
check this solution.
Check. Since this is a rational equation, we must check the solution
to make sure that it will not give us any zero denominators. The
checks to these types of exercises are very interesting and give the
student valuable experience with arithmetic with algebraic
Copy down the original equation
except that wherever you see an x, substitute the solution in
We need to find common denominators for the addition of
fractions problems on the top and bottoms of the two sides of the
After we cancel out the like terms we find we are left with
Invert and multiply
After the d + a - c - b cancels from the tops and bottoms on both
sides, it simplifies to
While both sides have generally the same format, they look
quite a bit different. But that can happen with equivalent rational
expressions. There are two ways to tell if two rational expressions
are equivalent. One is to cross multiply. The reader is invited to do
that and do verify that you get the same thing on both sides. The
other way is to see if we can simplify either side. That involves
factoring and canceling. Four term polynomials such as we find on
the top and bottom on both sides can sometimes be factored into a
product of two binomials. A necessary and sufficient condition for a
four term polynomial to be factorable into a product of two binomials
is that the product of the first and last terms be the same as the
product of the middle two terms. That does happen in the bottom of
the left side and the top of the right side, but it does not happen in
the other two expressions. So we look to see if it would be possible to
accomplish that with a rearrangement of the terms. If the products
of the first and last terms is the same as the product of the middle
two terms, they must at least have the same signs. That suggests that
one possible way to rearrange the terms is to put the positive terms
first and last and the negative terms together in the middle.
This will factor as
One advantage of factoring is that we can now tell when we
will get zero denominators. We will get zero denominators if
If a = b, the left side of the equation reduces down to 1, so if
the right side does not also reduce down to 1, which will happen
when c = d, there will be no solution. If both a = b and c = d, then
both sides reduce down to 1 for all x except b or d, and any such x
will be a solution.
If b = d, then the two equations have the same bottoms, in
which case the tops would have to be the same, which will happen
when a = c, in which case any x, except for b or d, would be a
Note that the left side reduces.
However, it will require a little more effort to reduce the right hand
side. Note that the d - c on the top is the negative of the c - d on
the bottom. These can be made to cancel by factoring a negative out
of one of them. Since the other factor on the top is the same as the
surviving factor on the top of the left, and the other factor on the
bottom of the right is the negative of the surviving factor on the
bottom of the left, factor a negative out of the c - d on the bottom of
and the solution checks.
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