Algebra

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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 $x$ , that you are finding is in its singular form. This means getting rid of squares or square roots attached to it.

Have a look at $p^2 = \dfrac{h}{x}$

Make $x$ the subject

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

Look again, and make $p$ the subject

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

Since $p$ is squared, we need to square root both sides

$p=\sqrt{\dfrac{h}{x}}$

Remember that there are 2 solutions for roots

Therefore $p=\pm\sqrt{\dfrac{h}{x}}$

Make c the subject of $a^2 = \dfrac{c}{d}$

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 $d$ the subject of $a^2 = \dfrac{c}{d}$

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 $p$ the subject of $x = \sqrt{\dfrac{h}{p}}$ ?

Square both sides to remove the square root

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

Switch $x^2$ and $p$

${\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 $\sqrt{x + c} = d$