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Relation between Roots and Coefficients for Quadratic

The equation ...

Question

The equation $a x^{2} + b x + c = 0$ where a, b, c are real numbers connected by the relation 4a+2b+c= 0 and ab > 0 has real roots.If you think this is true write 1 otherwise write 0 ?

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Solution

$Δ b^{2} 16 a^{2} + 4 a (4 a + 2 b)$ by given relation $= b^{2} + 16 a^{2} + 8 a b = + i v e$
Hence both roots are real. Alt.x=2 is a real root as $4 a + 2 b + c = 0$
and since imaginary roots occur in pairs therefore the other root must also be real.

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Similar questions

a, b,

c

be real numbers such that

4 a + 2 b + c = 0

a b > 0

. Then the equation

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

a, b, c, x

are all real numbers and

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

a, b, c

are in G.P., and

x

is their common ratio.If you think this is true write 1 otherwise write 0 ?

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

a, b, c

are real numbers,

a \neq 0

, be a quadratic equation, then this equation has no real roots if and only if

a, b, c \in R

4 a + 2 b + c = 0

a b > 0

.
Then the equation

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

Q. If the equations

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

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

have a common root, then their other roots are the roots of the equation,

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

If you think this is true write 1 otherwise write 0 .

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