# Quadratic Equation

If the equation is in the form of a*x*

^{2}+ b

*x*+ c= 0 (where a Â¹ 0). Then this known as quadratic equation or equation of degree 2.

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If the quadratic equation is a*x*^{2 }+ b*x* + c = 0

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Then the values of *x* are called roots

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and, Â Â Â Â Â Â

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*i.e.Â Â Â Â Â Â Â Â Â *

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Thus, the roots of the equation are :

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Symbolically, the roots can be expressed as â€“

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^{}

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# RelationÂ between Roots

Â Let one root be Â and the other root be b of the equation a*x*

^{2}+ b

*x*+ c = 0. Then

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Â Â Â Â Â Â Â Â Â Â Â Â Â Â

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Â Â Â Â Â Â Â Â Â Â Â Â Â Product of the roots Â =

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**Â Â **In order to find related Âµ and b, the following results may be useful :

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Â Â Â Â Â Â Â Â Â Â Â **Â Â (i) **Â Âµ^{2} + b^{2} = (Âµ + b)^{2} - 2Âµ b

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Â Â Â Â Â Â Â Â **Â Â Â Â (ii)**Â (Âµ - b)^{2} = (Âµ + b)^{2} - 4Âµ b

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Â Â Â Â Â Â Â Â Â Â Â **Â Â (iii) **Âµ^{3} + b^{3} = (Âµ + b)^{3} - 3Âµ b (Âµ + b)

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Â Â Â Â Â Â Â Â Â Â **Â Â Â (iv) **Âµ^{3} - b^{3}Â = (Âµ - b)^{3} + 3Âµ b (Âµ - b)

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# Construction of a quadratic equationÂ

ÂIf Âµ and b be the roots of the equation. Then the quadratic equation are :

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Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

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*Â Â Â i.e.Â Â Â Â Â *Â *x*^{2} â€“ (sum of the roots) *x* + Product of the roots = 0

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# Nature of the rootsÂ

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**1.**Â If b^{2} â€“ 4ac (discriminant) = 0; roots are real and equal

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**2.Â **If b^{2} â€“ 4ac < 0; roots are imaginary

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**3.Â **If b^{2} â€“ 4ac > 0 and a perfect square. Roots are real, rational and unequal.

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**4.Â **If b^{2} â€“ 4ac > 0 but not a perfect square. Roots are real, irrational and unequal.

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**Â Â Â Â Â Â Â **

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