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Introduction: Symmetrical Properties of Roots

The general form of a quadratic equation is \(ax^2+bx+c=0\), where \(a,b\) and \(c\) are constants and \(a\neq 0\). Dividing by \(a\) throughout, we get


If the roots of the equation are \(\alpha\) and \(\beta\), we can then write the equation with roots \(\alpha\) and \(\beta\) in the form

\[x=\alpha \quad \text{or} \quad x=\beta\]

\[\implies (x-\alpha)(x-\beta)=0\]

\[\implies x^2-(\alpha+\beta)x+\alpha\beta=0\cdots(2)\]

Comparing \((1)\) and \((2)\), we see that

\[\text{Sum of roots}=\alpha+\beta=-\frac{b}{a}\]

\[\text{Product of roots}=\alpha\beta=\frac{c}{a}\]

Note by Victor Loh
3 years ago

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Nice set! The problems are hard but managed to solve them all. Daniel Lim · 3 years ago

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Great note @Victor Loh Mardokay Mosazghi · 3 years ago

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Thank you :) @Mardokay Mosazghi Victor Loh · 3 years ago

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