Consider the equation
(
x
−
a
)
2
−
b
=
0
,
where
a
and
b
are both positive real numbers.
Which statement about the solution or solutions to the equation must be true?
A.
The equation has no real solutions.
B.
The equation has exactly one real solution and the solution is positive.
C.
The equation has exactly two real solutions and both of the solutions are positive.
D.
The equation has exactly two real solutions and at least one of the solutions is positive.
First, let's rearrange the equation by adding b to both sides:
(x − a)² = b.
Next, let's take the square root of both sides to solve for x:
x − a = ± √b.
To find x, we can add a to both sides:
x = a ± √b.
Since a and b are both positive real numbers, a is positive and the square root of b (√b) is positive as well.
Thus, the equation has exactly two real solutions (x = a + √b and x = a − √b) and at least one of the solutions is positive.
Therefore, the correct answer is choice D.