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Changing the Subject: Squares and Roots

Changing the Subject: Squares and Roots

Changing the Subject: Squares and Roots

Changing the subject of a formula sometimes involves manipulating squares and roots.

Some formulae involve squared terms and square root terms that need to be rearranged.

The most important thing here is to make sure that the variable, such as xx, that you are finding is in its singular form. This means getting rid of squares or square roots attached to it.

Have a look atp2=hxp^2 = \dfrac{h}{x}


Make xx the subject

xx is singular, so we can rearrange to x=hp2x=\dfrac{h}{p^2}


Look again, and make pp the subject

pp is squared, so we need an extra step to find the singular version of pp.


Since pp is squared, we need to square root both sides



Remember that there are 2 solutions for roots

Therefore p=±hxp=\pm\sqrt{\dfrac{h}{x}}

Make c the subject of a2=cda^2 = \dfrac{c}{d}

It is important to write ±  \pm \sqrt{\space\space} as, in a purely algebraic sense, you need to show that you know that there can be a positive square root as well as a negative one.

Make dd the subject of a2=cda^2 = \dfrac{c}{d}

We can also have formulae in which we need to remove a square root to isolate a term. The process here is the opposite of removing a squared term.

How can we make pp the subject of x=hpx = \sqrt{\dfrac{h}{p}}?


Square both sides to remove the square root

x=hpx = \sqrt{\dfrac{h}{p}} so x2=hpx^2=\dfrac{h}{p}


Switch x2x^2 and pp

{\color{#1fc49b}x^2} = \dfrac{h}{\color{#21affb}p}$${\color{#1fc49b}x^2} = \dfrac{h}{\color{#21affb}p}\rightarrow$${\color{#1fc49b}x^2} = \dfrac{h}{\color{#21affb}p}\rightarrow{\color{#21affb}p} = \dfrac{h}{\color{#1fc49b}x^2}

Make x the subject of x+c=d\sqrt{x + c} = d