If a - b - c -0- then the roots of the equation c 2- ab x 2-2 a 2- bc x - b 2- ac -0 areA- real and equalB- imaginaryC- real and unequalD- none of these

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Quadratic Equations

If a + b + c ...

Question

If a+b+c=0, then the roots of the equation $(c^{2} - a b) x^{2} - 2 (a^{2} - b c) x + (b^{2} - a c) = 0$ are

real and equal

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imaginary

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real and unequal

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none of these

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Solution

Finding D
$D = B^{2} - 4 A C$
$= 4 (a^{2} - b c)^{2} - 4 (b^{2} - a c) (c^{2} - a b)$
$= 4 [a^{4} + / b^{2} c^{2} - 2 a^{2} b c - / b^{2} c^{2} + a b^{3} - a^{2} b c + a c^{3}$
$= 4 [a (a^{3} + b^{3} + c^{3} - 3 a b c)]$
$= 4 [a (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a)]$
=0
So roots are real and equal.

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Similar questions

Q. The roots of the equation

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

are equal then

If the roots of the equation $(c^{2} - a b) x^{2} - 2 (a^{2} - b c) x + (b^{2} - a c) = 0$ are real ane equal, show that either $a = 0 o r (a^{3} + b^{3} + c^{3}) = 3 a b c$ .

Q. If the roots of the equation

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

are equal, then show that either a=0 or

a^{3} + b^{3} + c^{3} = 3 a b c

Q. If

c^{2} \neq a b

, and the root of

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

are equal then show that

a^{3} + b^{3} + c^{3} = 3 a b c

or a =0

Q. In the equation

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

it is given that

D = (b^{2} - 4 a c) > 0 .

Then, the roots of the equation are
(a) real and equal
(b) real and unequal
(c) imaginary
(d) none of these

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If a+b+c=0, then the roots of the equation (c2−ab)x2−2(a2−bc)x+(b2−ac)=0 are

Finding D D=B2−4AC =4(a2−bc)2−4(b2−ac)(c2−ab) =4[a4+/b2c2−2a2bc−/b2c2+ab3−a2bc+ac3 =4[a(a3+b3+c3−3abc)] =4[a(a+b+c)(a2+b2+c2−ab−bc−ca)] =0 So roots are real and equal.

If a+b+c=0, then the roots of the equation $(c^{2} - a b) x^{2} - 2 (a^{2} - b c) x + (b^{2} - a c) = 0$ are