Substitution Worked ExamplesIGCSE Maths 0580

Substitute first, simplify second. Rearranging the formula before substituting is extra work and a common source of sign errors.

Only $t$ is squared, not $at$. Writing $\tfrac{1}{2}(-10 \times 3)^2$ gives $450$ and loses every mark — the power attaches to $t$ alone.

Complete the top and bottom fully before dividing. Cancelling an $a$ from numerator and denominator is invalid here — you may only cancel factors of the whole expression, not individual terms.

$(-3)^2 = 9$, but $-3^2 = -9$. Without the bracket the answer becomes $-3$. This is a real error: the June 2025 examiner report found that when a table of values was completed, the square of $-1$ was sometimes worked out as $-1$ rather than $+1$. Write the bracket every time you substitute a negative value.

In (b) the squaring is what matters — doubling $d$ does not halve $P$. Checking with numbers confirms it: at $d = 10$, $P = \frac{200}{100} = 2$, which is a quarter of $8$.

Substitute first and solve the resulting equation — rearranging the formula for $b$ before substituting works but takes longer and risks an error. The March 2025 examiner report advised candidates "to check their answer makes sense for the context of the problem": substituting $b = 9$ back costs one line and confirms the result.

Substituting $h = 40$ straight from the diagram gives $1650$, and the accuracy mark is lost for mixing units. The formula has only one $lw$ term because the box has a base but no lid. On the non-calculator paper, the March 2025 examiner report said candidates "need to focus on efficient arithmetic strategies": $8 \times 19.5$ is easiest as $8 \times 20 - 8 \times 0.5 = 156$.

Using the diameters $30$ and $20$ as $R$ and $r$ makes every term in the bracket four times too big, giving about $47.8$ litres. The other common slip is leaving the answer in $\text{cm}^3$ when the question asks for litres. In (b), keep $h$ as the unknown and solve. The June 2025 examiner report stressed that "intermediate values within longer methods should not be rounded", so keep $3800\pi$ exact and round only the final answer.

Both parts of (a) and (b)(i) are 'show that' questions. The March 2026 examiner report said that candidates "should show every step in their method", so write the substitution out before simplifying. A common slip when clearing fractions in (b)(i) is to leave the $25$ on the left unmultiplied — it must also be multiplied by $2h$. In (ii), state the reason in words: an arch of this shape cannot be taller than its radius.

The March 2026 examiner report noted that a few candidates wrote the square of a negative number without a bracket and so used a negative value in their evaluation. Here that gives $k^2 = -49 + 27 = -22$, which has no square root — a clear sign something has gone wrong. On a calculator, type the bracket: $(-7)^2$, not $-7^2$. Stopping at $k^2 = 76$ loses the final mark.

In (a), the energy cannot be negative: the March 2025 examiner report found that errors came from forgetting to square the negative value or squaring it incorrectly, which led to a negative answer. In (b), the same report noted that "some candidates forgot to square root", stopping at $v^2 = 100$. The answer line asks for $v$, not $v^2$.