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Nature of Roots

If the roots ...

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If the roots α and β of the equation ax^2+bx+c=0 are real and of opposite sign then the roots of the equation α(x-β)^2 +β(x-α)^2 is/are (1) positive (2) Negative (3) Real and opposite (4) imaginary

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α, β

are roots of equation

a x^{2} + b x + c = 0

which are real and opposite in sign, then roots of the equation

α {(x - β)}^{2} + β {(x - α)}^{2} = 0

α

β

are roots of the quadratic equation

a x^{2} + b x + c = 0

. The equation has real roots which are of opposite signs, then the equation

α (x - β)^{2} + β (x - α)^{2} = 0

α

β

are the roots of

a x^{2} + b x + c = 0 (b \neq 0)

α β < 0.

Then the roots of

α (x - β)^{2} + β (x - α)^{2} = 0

α

β

a x^{2} + b x + c = 0

then the equation whose roots are

α^{2}

β^{2}

α, β

are the roots of the equation

a x^{2} + b x + c = 0

log (a - b x + c x^{2}) = log a + (α + β) x - \frac{α^{2} + β^{2}}{2} x^{2} + \frac{α^{3} + β^{3}}{3} x^{3} - . . .

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