Changing the subject with roots and products

Harder rearrangements: subject in a product, under a root, or appearing twice.

Key formulas

\text{Collect terms in the subject}, \text{then factor}

\text{Undo squares with} \pm \sqrt (\text{context decides sign})

\text{Clear fractions before isolating}

If s appears on both sides or in two terms, bring those terms together and factor s out: as + b = cs → as − cs = −b → s(a − c) = −b.

From A = πr², r = √(A/π) for r > 0. Squaring both sides of an equation can introduce extras — check in the original.

Physics-style formulas (v = u + at, s = ut + ½at²) appear in maths papers as pure rearrangement. Ignore the science story; follow inverse operations.

State the subject on the left as s = … and refuse to leave compound fractions unsimplified.

Problem. Make t the subject of v = u + at.

Solution. v − u = at ⇒ t = (v − u)/a.

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v=u+at ⇒ t=

When undoing an even power and both signs are mathematically possible; lengths stay positive.

Yes — equality is symmetric.