Question

X13.) Consider the equation a√x-2, where a is a real number. A student claims for all

possible values of a, the equation will have exactly 1 real solution. Do you agree or
disagree with the student's claim that all possible values of a will result in an equation
with exactly 1 real solution? Justify your reasoning.

Answer 1

I disagree with the student's claim that all possible values of a will result in an equation with exactly 1 real solution.

Explanation:

The number of real solutions to the given equation depends on the value of a and x, and not all values of a will result in exactly one real solution.

To see why, let us consider the discriminant of the equation: (a√x-2)^2 = a^2(x-4). For the equation to have exactly one real solution, the discriminant must be equal to zero. This gives us the equation a^2(x-4) = 0, which has a unique real solution x = 4 for a ≠ 0. However, when a = 0, the equation reduces to -2 = 0, which has no real solutions.

Therefore, the equation a√x-2 has exactly one real solution for all possible values of a except for a = 0, where the equation has no real solutions. Hence, the student's claim is incorrect, and the number of real solutions to the equation depends on the value of a.