Let m and n be the numbers of real roots of the quadratic equations x 2 - 12 x + [ x ] + 31 = 0 and x 2 - 5 | x + 2 | - 4 = 0 respectively, where [ x ] denotes the greatest integer ? x . Then m 2 + m n + n 2 is equal to __________ .

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Solution

Q.
Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^{2} - 12 x + [x] + 31 = 0$ and $x^{2} - 5$ $| x + 2 |$ $- 4 = 0$ respectively, where $[x]$ denotes the greatest integer $\leq x$ . Then $m^{2} + m n + n^{2}$ is equal to __________ . [2023]

Ans.
(9)

$x^{2} - 12 x + [x] + 31 = 0$

$\Rightarrow {x} = x^{2} - 11 x + 31 \Rightarrow 0 \leq x^{2} - 11 x + 31 < 1$

$\Rightarrow x^{2} - 11 x + 30 < 0 \Rightarrow x \in (5, 6)$

$So, [x] = 5$

$Now, x^{2} - 12 x + 5 + 31 = 0$

$\Rightarrow x^{2} - 12 x + 36 = 0 \Rightarrow x = 6 but x \in (5, 6)$

$m = 0$

$Now, for x^{2} - 5$ $| x + 2 |$ $- 4 = 0$

When

$x \geq - 2;$ $x < - 2$

$x^{2} - 5 x - 14 = 0;$ $x^{2} + 5 x + 6 = 0$

$(x - 7) (x + 2) = 0;$ $(x + 3) (x + 2) = 0$

$x = 7, - 2;$ $x = - 3, - 2$

$Now, x^{2} - 5$ $| x + 2 |$ $- 4 = 0 \Rightarrow x = {7, - 2, - 3}$ $∴ n = 3$

$So, m^{2} + m n + n^{2} = 9$

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