First slide

Theory of expressions

Question

The number of integral values of m for which the equation $(1 + m^{2}) x^{2} - 2 (1 + 3 m) x + (1 + 8 m) = 0$ has no real roots is

Moderate

A

1
B

2
C

3
D

infinitely many

Solution

Let D be the discriminant of the given equation
Then,
$\begin{array}{l} D = 4 (1 + 3 m)^{2} - 4 (1 + m^{2}) (1 + 8 m) \\ \Rightarrow D = 4 \{9 m^{2} + 6 m + 1 - 8 m^{3} - m^{2} - 8 m - 1\} \\ \Rightarrow D = 4 (- 8 m^{3} + 8 m^{2} - 2 m) \\ \Rightarrow D = - 8 m (4 m^{2} - 4 m + 1) = - 8 m (2 m - 1)^{2} \end{array}$
If the given equation has no real roots, then
$D < 0 \Rightarrow - 8 m (2 m - 1)^{2} < 0 \Rightarrow m > 0 and m \neq \frac{1}{2}$
Hence, m can take infinitely many integral values.

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