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Relation between Roots and Coefficients for Quadratic

If the equati...

Question

If the equations $a x^{2} + b x + c = 0$ and $x^{2} + 2 x + 4 = 0$ have a common root then find a : b : c.

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Solution

$9 x^{2} + b x + c = 0$
$x^{2} + 2 x + 4 = 0$
Let $D < 0$ , so it have both root common
$\frac{a}{1} = \frac{b}{2} = \frac{c}{4}$
$a : b : c = 1 : 2 : 4$

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Q. If the equations

x^{2} + 2 x + 3 = 0

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

, have a common root, then

a : b : c

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

x^{2} + 2 x + 3 = 0

have a common root, then if

a : b : c = A : B : C

A + B + C =

Q. Assertion :If the equation

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

x^{2} + 2 x + 3 = 0

have a common root , then

a : b : c

1 : 2 : 3

. Reason: The roots of the equation

x^{2} + 2 x + 3 = 0

a, b, c ϵ R

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

x^{2} + 2 x + 9 = 0

have a common root, show that

a : b : c = 1 : 2 : 9.

Q. Assertion :If equation

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

x^{2} - 3 x + 4 = 0

have exactly one root common, then at least one of

a, b, c

is imaginary. Reason: If

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are not all real, then equation

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

can have one real root and one one root imaginary.

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