National 5MathsSQARevision NotesAlgebraic SkillsChanging the SubjectFormulae with Squares & Roots

Formulae with Squares & Roots (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

Changing the subject with squares and roots

How do I change the subject when the new subject is squared?

  • E.g., make v the subject of E equals 1 half m v squared

  • Start by following the usual algebraic steps to get v squared alone on one side of the equation

    • Multiply both sides by 2

      • 2 E equals m v squared

    • Divide both sides by m

      • fraction numerator 2 E over denominator m end fraction equals v squared

  • Squaring and taking a square root are inverse (i.e. opposite) operations

    • So take the square root of both sides to get rid of the square on the v

      • v equals plus-or-minus square root of fraction numerator 2 E over denominator m end fraction end root

    • You will need additional information to decide whether to use the positive square root or the negative square root (see the Worked Example)

Examiner Tips and Tricks

Don't forget to consider the plus-or-minus in cases like this.

This is because negative numbers and positive numbers both square to give a positive number. For example:

x squared equals 4 space space rightwards double arrow space space x equals plus-or-minus square root of 4 space space rightwards double arrow space space x equals space 2 space or space minus 2

How do I change the subject when the new subject is inside a square root?

  • E.g., make l the subject of T equals 2 pi square root of l over g end root

  • Start by following the usual algebraic steps to get square root of l over g end root alone on one side of the equation

    • Divide both sides by 2 pi

      • fraction numerator T over denominator 2 pi end fraction equals square root of l over g end root

  • Squaring and taking a square root are inverse (i.e. opposite) operations

    • So square both sides to get rid of the square on the right hand side

      • open parentheses fraction numerator T over denominator 2 pi end fraction close parentheses squared equals l over g

    • Then multiply both sides by g to get l on its own

      • l equals g open parentheses fraction numerator T over denominator 2 pi end fraction close parentheses squared

      • You can also expand the brackets to get l equals fraction numerator g T squared over denominator 4 pi squared end fraction

What if the square or square root appears on other parts of the formula?

  • Only try to remove squares or square roots if you need to do so to release the new subject

    • Otherwise a square (or other power) or square root can be moved around the equation 'as is'

    • This is similar to dealing with brackets when rearranging formulas

  • E.g., make m the subject of E equals 1 half m v squared

    • Multiply both sides by 2

      • 2 E equals m v squared

    • Divide both sides by v squared

      • m equals fraction numerator 2 E over denominator v squared end fraction

Worked Example

(a) Change the subject of the formula space y equals a square root of x plus b to x.

(b) Change the subject of the formula B equals 1 third m n squared plus 5 n to m.

Answer:

Part (a)

Subtract b from both sides to get a square root of x alone on the right hand side

space y minus b equals a square root of x

Divide both sides by a to get square root of x on its own

space fraction numerator y minus b over denominator a end fraction equals square root of x

Square both sides to get rid of the square root

  • Be sure to square everything on the left hand side by putting it all in brackets first

space open parentheses fraction numerator y minus b over denominator a end fraction close parentheses squared equals open parentheses square root of x close parentheses squared

x equals open parentheses fraction numerator y minus b over denominator a end fraction close parentheses squared
 

Part (b)

Subtract 5 n from both sides

B minus 5 n equals 1 third m n squared

Multiply both sides by 3 to get rid of the fraction

  • Put everything on the left hand side in brackets first

3 open parentheses B minus 5 n close parentheses equals m n squared

Divide both sides by n squared to get m on its own

  • The 'squared' is not on the new subject m, so there's no other tricky steps to worry about here!

fraction numerator 3 open parentheses B minus 5 n close parentheses over denominator n squared end fraction equals m

m equals fraction numerator 3 open parentheses B minus 5 n close parentheses over denominator n squared end fraction space space or space space m equals fraction numerator 3 B minus 15 n over denominator n squared end fraction

Worked Example

The area of a circle is given by the formula

A equals pi r squared

where r is the radius of the circle.

Make r the subject of the formula.

Answer:

Divide both sides by pi to get r squared on its own

space A over pi equals r squared

Take the square root of both sides to get rid of the square on the r

  • Put everything on the left hand side in a single square root

  • Remember that taking a square root gives a plus-or-minus answer

plus-or-minus space square root of A over pi end root equals r

Here r is the radius of a circle, which cannot be negative

  • So you only need the positive answer

r equals square root of A over pi end root

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.