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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

$x^2+\frac{b}{a}x+\frac{c}{a}=0\cdots(1)$

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. · 3 years ago

Great note @Victor Loh · 3 years ago