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Q.
If $a + b + c = 0,$ then the quadratic equation $3ax^2 + 2bx + c = 0$ has
Application of Derivatives
Solution:
Let a + b + c = 0
Consider function $f (x) = ax^3 + bx^2 + cx$ which is continuous and differentiable being a polynomial.
Now, f (0) = 0, f (1) = a + b + c = 0 (given)
$\therefore $ By rolle’s theorem a point $\alpha \in (0, 1)$ such that f ' ($\alpha$) = 0
$ \Rightarrow \, 3a \alpha^2 + 2b \alpha + c = 0$
$\Rightarrow $ a is root of $3ax^2 + 2bx + c = 0$