2. Solve for   x   and check.

First clear denominators. In this case that amounts to cross multiplying.

(x + a)(x + d) = (x + b)(x + c)

Now that we have gotten rid of denominators, we will want to simplify both sides. Remove parentheses.

x2 + dx + ax + ad = x2 + cx + bx + bc

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   x2   from both sides,

dx + ax + ad = cx + bx + bc

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.

dx + ax - cx - bx = bc - ad

At this point, all of the terms on the left side have a factor of   x,   so we can factor it out.

(d + a - c - b)x = bc - ad

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.

Multiply the tops, and add the fractions with the common denominators.

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 the fractions. 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

• a = b,
• b = d, or
• c = d

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.

When we cancel as indicated above, we have a couple of factors where we are subtracting two terms in the reverse order. We have seen that this reduces to   -1,   so we get.

and the solution checks.