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

Step-by-step solution

Given, x 2 - 12 x + x + 31 = 0 ⇒ x 2 - 12 x + 31 ⏟ ≥ - 5 = - x Now from above equation we say that, it could have its solution in [ 5 , 6 ) but it does not exist as at x = 5 as LHS = 1 , So no solution, hence m = 0 Now solving, x 2 - 5 x + 2 - 4 = 0 Taking Case 1 when x ≥ - 2 we get, x 2 - 5 x + 2 - 4 = 0 ⇒ x 2 - 5 x - 14 = 0 ⇒ x = 7 , - 2 Now taking Case 2 when x < - 2 we get, x 2 + 5 x + 10 - 4 = 0 ⇒ x = - 2 , - 3 So, total 3 solution i.e., x = - 3 ,   - 2 , &#16