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

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 $E=\frac{1}{2}m{v}^2$

• Start by following the usual algebraic steps to get $v^2$ alone on one side of the equation

  • Multiply both sides by 2

    • $2E=m{v}^2$

  • Divide both sides by $m$

    • $\frac{2E}{m}=v^2$

• 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 $v$

    • $v=\pm \sqrt{\frac{2E}{m}}$

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

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

• E.g., make $l$ the subject of $T=2\pi \sqrt{\frac{l}{g}}$

• Start by following the usual algebraic steps to get $\sqrt{\frac{l}{g}}$ alone on one side of the equation

  • Divide both sides by $2\pi$

    • $\frac{T}{2\pi }=\sqrt{\frac{l}{g}}$

• 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

    • ${\left(\frac{T}{2\pi }\right)}^2=\frac{l}{g}$

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

    • $l=g{\left(\frac{T}{2\pi }\right)}^2$

    • You can also expand the brackets to get $l=\frac{g{T}^2}{4{\pi }^2}$

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 $E=\frac{1}{2}m{v}^2$

  • Multiply both sides by 2

    • $2E=m{v}^2$

  • Divide both sides by $v^2$

    • $m=\frac{2E}{v^2}$