Question

The number of integers $n$ for which $3x^2 - 25x + n = 0$ has three real roots is

• A. 1
• B. 25
• C. 55
• D. infinite

Answer 1

Answer: (C)

We have, $3x^{2} - 25x + n = 0$
Let $ f\left(x\right) = 3x^{3} - 25x + n $
$f'\left(x\right) = 9x^{2} - 25 $
Put $f'\left(x \right) = 0$
$9x^{2} -25 = 0 $
$x = \pm \frac{5}{3} $
$\Rightarrow x_{1} = \frac{-5}{3}, x_{2} = \frac{5}{3} $
$f(x) $ has three real roots.
$\therefore f(x_1) f(x_2) < 0$
$\therefore \left(3 \left( \frac{-5}{3}\right)^{3} -25\left(\frac{-5}{3}\right) + n\right)$
$\left(3\left(\frac{5}{3}\right)^{3} - 25\left(\frac{5}{3}\right) + n\right) < 0$
$\left(\frac{-125}{9} + \frac{125}{3} +n\right)\left( \frac{125}{9} - \frac{125}{3} +n\right) < 0$
$ \left( n + \frac{250}{9}\right)\left(n - \frac{250}{9}\right) < 0$
$n \in \left( \frac{-250}{9} , \frac{250}{9}\right)$
Hence, $n \in I$
$\therefore $ Total integer is $55$.