Fractions and Valid Roots
Some equations become quadratic only after fractions are cleared or a context is translated into algebra. Record restrictions before solving, then test every root against the original problem.
Equations with fractions in them
- An equation with on the bottom of a fraction often turns into a quadratic once you clear the fractions.
- Multiply every term by the product of the bottoms. That clears them all in one step.
- A denominator is the bottom of a fraction. First write the values that make any denominator zero. Those values are not allowed in the original equation and must never appear in the final answer.
Worked example
Solve
- The original equation requires and . The bottoms are and . Multiply every term by .
The left side becomes
The right side becomes
- Expand both sides.
- Collect everything on one side.
- Factorise and solve.
- Neither root is excluded by . Substituting in the original equation confirms both.
Answer
Your turn9709/11/O/N/22 Q1
Solve .
Show worked answer
The original equation requires . Multiply both sides by .
Now factorise and solve as usual.
Answer
Rejecting invalid roots
- Sometimes the question limits what can be. A length cannot be negative, for example.
- The algebra does not know about that limit. It gives you both roots, and you must reject the one that breaks the rule.
Worked example
A rectangle with a given area
A rectangle has sides of length cm and cm. Its area is cm². Find the lengths of the sides.
- Write the area as an equation, then bring it to zero.
- Factorise and solve.
- Reject the negative root, because is a length. So , and the other side is .
Answer
The sides are 4.5 cm and 20 cm.
- Always read the question again before you write your final answer. Ask yourself whether both roots are allowed.
This same finish appears later in series, trigonometry, coordinate geometry and hidden-quadratic questions: form a quadratic, solve it, then return to the original meaning and check which roots are valid.