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Consider two quadratic expressions $f(x) = ax^{2} + bx + c$ and $g(x) = ax^{2} + px + q$ , $(a, b, c, p, q \epsilon R, b \neq p )$ such that their discriminants are equal. If $f(x) = g(x)$ has a root $x = \alpha$ , then
Option: 1
$\alpha$ will be A.M. of the roots of $f(x) = 0$ and $g(x) = 0$

Option: 2
$\alpha$ will be A.M. of the roots of $f(x) = 0$

Option: 3
$\alpha$ will be A.M. of the roots of $f(x) = 0$ or $g(x) = 0$

Option: 4
$\alpha$ will be A.M. of the roots of $g(x) = 0$

Answers (1)

best_answer

Discriminant of Quadratic Equation -

$D=b^{2}-4ac$

- wherein

$ax^{2}+bx+c= 0$

is the quadratic equation

$a\alpha^{2}+b \alpha + c=a\alpha^{2}+p\alpha+q$

$\therefore \alpha =\frac{q-c}{b-p}$

A.M. of roots of $f(x) = 0$ and $g(x) = 0$ is

= $\frac{-b/a-p/a}{4}=-\frac{b+p}{4a}$ (Number of roots is 4)

Since discriminants are equal

$b^{2}-4ac=p^{2}-4aq$

$b^{2}-p^{2}=4ac-4aq$

$\frac{b+p}{4a}=\frac{c-q}{b-p}=-\alpha$

$\therefore \alpha$ =A.M. of roots of f(x)=0 and g(x)=0

Posted by

Gaurav

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