TMUA 2021 · Paper 2 · Question 11

A student attempts to solve the following problem, where $a$ and $b$ are non-zero real numbers: > Show that *if* $a^{2} - 4 b^{3} \geq 0$ *then* there exist real numbers $x$ and $y$ such that $a = x y (x + y)$ and $b = x y$ .

Consider the following attempt:

(x - y)^{2} \geq 0 (I)

so x^{2} + y^{2} - 2 x y \geq 0 (II)

so (x + y)^{2} - 4 x y \geq 0 (III)

so x^{2} y^{2} (x + y)^{2} - 4 x^{3} y^{3} \geq 0 (IV)

so a^{2} - 4 b^{3} \geq 0 (V)

Which of the following best describes this attempt?

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