2. Solve for x and check.

 

First clear denominators. In this case that amounts to cross

multiplying.

 

 

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

expressions.

 

Copy down the original equation

 

 

 

 

except that wherever you see an x, substitute the solution in

parentheses.

 

 

 

 

We need to find common denominators for the addition of

fractions problems on the top and bottoms of the two sides of the

equation.

 

 

 

 

Remove parentheses

 

 

 

 

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

solution.

 

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

the right.

 

 

 

 

 

 

 

and the solution checks.

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