Rational Equations
Some rational equations (an equation with one or more fractions as terms) become quadratic equations once each term has been multiplied by the least common denominator. Remember, you must be sure that any solutions do not lead to a zero in any denominator in the original equation.
There are two main approaches to clearing
the denominator(s) in a rational equation. If the equation is in the form of
‘‘fraction = fraction,’’ cross multiply. If the equation is not in this form,
find the least common denominator (LCD). Finding the least common denominator
often means you need to factor each denominator completely. We learned in
Chapter 7 to multiply both sides of the LCD, then to distribute the LCD. In
this chapter we will simply multiply each term on each side of the equation by
the LCD. If the new equation is a quadratic equation, collect all terms on one
side of the equal sign and leave a zero on the other. In the examples and
practice problems below, the solutions that lead to a zero in a denominator
will be stated.
Examples
This is in
the form ‘‘fraction = fraction’’ so we will cross multiply.
First factor
each denominator, second find the LCD, and third multiply all three terms by
the LCD.
We cannot
let x be 4 because x ¼ 4 leads to a zero in the denominator of 
. The only solution is x = 2.
Practice
Because all
of these problems factor, factoring is used in the solutions. If factoring
takes too long on some of these, you may use the quadratic formula.
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Solutions
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