Standard +0.8 This question requires understanding of repeated roots, forming equations from roots, and algebraic manipulation across two related quadratics. Students must recognize that α = -p/2, then use the constraint m = n in the second equation to establish relationships, requiring several non-trivial steps and careful algebraic reasoning beyond standard textbook exercises.
3. The quadratic equation \(x ^ { 2 } + p x + q = 0\) has a repeated root \(\alpha\).
A new quadratic equation has a repeated root \(\frac { 1 } { \alpha }\) and is of the form \(x ^ { 2 } + m x + m = 0\).
Find the values of \(p\) and \(q\) in the original equation.
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3. The quadratic equation $x ^ { 2 } + p x + q = 0$ has a repeated root $\alpha$.
A new quadratic equation has a repeated root $\frac { 1 } { \alpha }$ and is of the form $x ^ { 2 } + m x + m = 0$.\\
Find the values of $p$ and $q$ in the original equation.\\
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\hfill \mbox{\textit{WJEC Further Unit 1 2024 Q3 [6]}}