If the roots of the equation x^2 + a^2 = 8x + 6a are real, then find the interval to which a belongs.

Given that roots of equation x^2 + a^2 = 8x + 6a are real.

x^2 + a^2 = 8x + 6a

=> x^2 - 8x + a^2 - 6a = 0

Since the roots of this equation are real, the discriminant of this equation must be >= 0

=> (-8)^2 - 4(1)(a^2 - 6a) >= 0

=> 64 - 4a^2 + 24a >= 0 

=> -4a^2 + 24a + 64 >= 0

=> 4a^2 - 24a - 64 <= 0

=> a^2 - 6a - 16 <= 0

=> a^2 + 2a - 8a - 16 <= 0

=> a(a + 2) - 8(a + 2) <= 0

=> (a - 8)(a + 2) <= 0

hence for (a - 8)(a + 2) to be <= 0, a should belong to the interval [-2, 8].

So, a belongs to [-2, 8]

*If you are new to inequalities, you may have to read our knowledge based article Solving Inequalities in order to understand how we solved this inequality.