Question

how to find the zeroes of the quadratic polynomial and verify the Relationship between zeroes and the coefficient
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Answer 1

How to find the zeros of the quadratic polynomial?

• FACTORING
• QUADRATIC FORMULA
• COMPLETING THE SQUARE

How to verify the Relation between zeroes and the coefficient?

1. QUADRATIC FORMULA

Let there be equation $ax^2+bx+c=0$.

By the quadratic formula,

$x=\frac{-b\pm \sqrt{b^2-4ac} }{2a}$.

Sum of two zeros : $-\frac{b}{a}$

∵ $\frac{-b +\sqrt{b^2-4ac} }{2a} + \frac{-b- \sqrt{b^2-4ac} }{2a} =\frac{-2b}{2a} =-\frac{b}{a}$

Multiplication of two zeros : $\frac{c}{a}$

∵$\frac{-b +\sqrt{b^2-4ac} }{2a} \times \frac{-b- \sqrt{b^2-4ac} }{2a}$

$=\frac{b^2-(b^2-4ac)}{4a^2} =\frac{4ac}{4a^2} =\frac{c}{a}$

2. IDENTITY

Let there be polynomial $ax^2+bx+c$,

and let their zeros be α and β.

Then the polynomial $ax^2+bx+c$ factors to be $a(x-\alpha )(x-\beta )$.

$ax^2+bx+c=a(x-\alpha )(x-\beta )$

$ax^2+bx+c=ax^2-a(\alpha +\beta)x+a\alpha\beta$

By the identity,

$-(\alpha +\beta )x=bx$,

and $a\alpha \beta =c$.

∴ $\alpha +\beta =-\frac{b}{a}$

∴ $\alpha \beta =\frac{c}{a}$

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