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Algebraic Fractions

Solving Algebraic Fractions

How do I solve an equation that contains algebraic fractions?

  • There are two methods for solving equations that contain algebraic fractions
  • One method is to deal with the algebraic fractions by adding or subtracting them first and then solving the equation 
    • Follow the rules for solving a linear equation containing a fraction on one or both sides
      • Remove the fractions first by multiplying both sides by everything on the denominator
      • Remember to put brackets around any expression that you multiply by
  • The second method is to begin by multiplying everything in the fraction by each of the expressions on the denominator
    • This will remove the denominators of the fractions, leaving you with either a linear or a quadratic equation to solve   
    • Multiplying everything in the fraction by the common denominator is a way of carrying out this process in one go
  • For example, to solve the equation fraction numerator 4 over denominator x space minus space 3 end fraction space plus space fraction numerator 5 over denominator x space plus space 1 end fraction space equals space 5 you will need to multiply every term in the equation by both open parentheses x space minus space 3 close parentheses and open parentheses x space plus space 1 close parentheses 
    • STEP 1
      Multiply every term by open parentheses x space minus space 3 close parentheses

table row cell fraction numerator 4 over denominator x space minus space 3 end fraction open parentheses x space minus space 3 close parentheses space plus space fraction numerator 5 over denominator x space plus space 1 end fraction open parentheses x space minus space 3 close parentheses space end cell equals cell space 5 open parentheses x space minus space 3 close parentheses end cell row cell fraction numerator 4 over denominator up diagonal strike open parentheses x space minus space 3 close parentheses end strike end fraction up diagonal strike open parentheses x space minus space 3 close parentheses end strike space plus space fraction numerator 5 open parentheses x space minus space 3 close parentheses space over denominator x space plus space 1 end fraction end cell equals cell space 5 open parentheses x space minus space 3 close parentheses end cell row cell 4 space plus space fraction numerator 5 open parentheses x space minus space 3 close parentheses space over denominator x space plus space 1 end fraction end cell equals cell space 5 open parentheses x space minus space 3 close parentheses end cell end table

    • STEP 2
      Multiply every term by table row blank blank cell open parentheses x space plus space 1 close parentheses end cell end table

table row cell 4 open parentheses x space plus space 1 close parentheses space plus space fraction numerator 5 open parentheses x space minus space 3 close parentheses over denominator x space plus space 1 end fraction open parentheses x space plus space 1 close parentheses end cell equals cell space 5 open parentheses x space minus space 3 close parentheses open parentheses x space plus space 1 close parentheses end cell row cell 4 open parentheses x space plus space 1 close parentheses space plus space fraction numerator 5 open parentheses x space minus space 3 close parentheses over denominator open parentheses up diagonal strike x space plus space 1 end strike close parentheses end fraction up diagonal strike open parentheses x space plus space 1 close parentheses end strike end cell equals cell space 5 open parentheses x space minus space 3 close parentheses open parentheses x space plus space 1 close parentheses end cell row cell 4 open parentheses x space plus space 1 close parentheses space plus space 5 open parentheses x space minus space 3 close parentheses space end cell equals cell space 5 open parentheses x space minus space 3 close parentheses open parentheses x space plus space 1 close parentheses end cell end table

    • STEP 3
      Expand the brackets on both sides and simplify

table row cell 4 x space plus space 4 space plus space 5 x space minus space 15 space end cell equals cell space 5 open parentheses x squared space plus space x space minus space 3 x space minus 3 close parentheses end cell row cell 9 x space minus space 11 space end cell equals cell space 5 open parentheses x squared space minus space 2 x space minus 3 close parentheses end cell row cell 9 x space minus space 11 space end cell equals cell space 5 x squared space minus space 10 x space minus 15 end cell end table

    • STEP 4
      Rearrange the equation so that it is in a form that can be solved

table row cell 9 x space minus space 11 space end cell equals cell 5 x squared space minus space 10 x space minus 15 end cell row cell negative open parentheses 9 x space minus space 11 close parentheses space space space space space space space space space space space space space space space space space space space space end cell equals cell space space space space space space space space space space space space space space space space space space space space minus open parentheses 9 x space minus space 11 close parentheses space space space space space end cell row cell 0 space end cell equals cell space 5 x squared space minus space 10 x space minus 9 x space minus space 15 space plus space 11 end cell row cell 0 space end cell equals cell space 5 x squared space minus space 19 x space minus space 4 end cell end table

    • STEP 5
      Solve the equation
      • You can swap the sides if it makes solving the equation easier    

    table row cell space 5 x squared space minus space 19 x space minus space 4 space end cell equals cell space 0 end cell row cell open parentheses 5 x space plus space 1 close parentheses open parentheses x space minus space 4 close parentheses space end cell equals cell space 0 end cell row cell x space end cell equals cell space minus 1 fifth space or space space x space equals space 4 end cell end table

Exam Tip

  • Multiplying by both denominators at once can speed up the process, but be careful with the algebra if choosing this technique

Worked example

fraction numerator 2 over denominator p plus 3 end fraction minus 5 over p equals 6 p

Show that this equation can be written as  6 p cubed plus 18 p squared plus 3 p space plus space 15 equals 0.

To clear the fractions, we multiply both sides by the denominators.
We can do this one denominator at a time. We can start by multiplying by left parenthesis p plus 3 right parenthesis.

2 minus fraction numerator 5 left parenthesis p plus 3 right parenthesis over denominator p end fraction equals 6 p left parenthesis p plus 3 right parenthesis

Now multiply by p

2 open parentheses p close parentheses space minus space 5 open parentheses p plus 3 close parentheses space equals space 6 p open parentheses p plus 3 close parentheses open parentheses p close parentheses space

Now expand brackets

2 p space minus space 5 open parentheses p plus 3 close parentheses space equals space 6 p squared open parentheses p plus 3 close parentheses 2 p space minus space 5 p minus 15 space equals space 6 p cubed plus 18 p squared

Collect like terms

negative 3 p minus 15 equals 6 p cubed plus 18 p squared


Add the terms on the left hand side to the right hand side, to complete the question

0 space equals space 6 p cubed plus 18 p squared plus 3 p plus 15

bold 6 bold italic p to the power of bold 3 bold plus bold 18 bold italic p bold plus bold 3 bold italic p bold plus bold 15 bold space bold equals bold space bold 0

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