The difference of the squares of two numbers is 180. The square of the smaller number is 8 times the greater number. Find the two numbers.
Generated by claude-sonnet-4-6 · 2026-06-15 10:33 · grounding rag
Model Answer
Let the greater number = $x$ and the smaller number = $y$.
Setting up equations:
Given: $x^2 - y^2 = 180$ … (1)
Given: $y^2 = 8x$ … (2)
Substituting (2) in (1):
$$x^2 - 8x = 180$$
$$x^2 - 8x - 180 = 0$$
Factorising:
$$x^2 - 18x + 10x - 180 = 0$$
$$x(x - 18) + 10(x - 18) = 0$$
$$(x + 10)(x - 18) = 0$$
So, $x = 18$ or $x = -10$.
Since $y^2 = 8x$ and $y^2$ must be non-negative, $x$ cannot be negative.
∴ $x = 18$
From (2): $y^2 = 8 \times 18 = 144 \Rightarrow y = \pm 12$
The two numbers are 18 and 12 (or 18 and −12).
Source: Chapter 4, Quadratic Equations
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Explanation
- Examiners award marks for: correct variable assignment, forming both equations, substituting to get the quadratic, correct factorisation, and finding both values of $y$ (±12).
- A common mistake is rejecting $x = -10$ without justification — always state why (here, $y^2 = 8x$ requires $x \geq 0$).
- Both $y = 12$ and $y = -12$ are valid answers; write both to be safe.