Two or more math objects are distinct if they are not the same object. The objects may be the same size and similar, or even congruent, and still be distinct. Mathematicians use the word distinct to emphasize that two or more math objects are not the same object.
For example, a quadratic equation may have two distinct complex roots, two distinct
real roots or two real roots that are the same.
x^{2} - x - 6 = 0 → (x + 2)(x - 3) = 0: x has 2 distinct real roots.
x^{2} + 2x + 2 = 0 → (x + 1)(x + 1) = 0: x does not have 2 distinct real roots.
x^{2} + 4 = 0 → (x + 2i)(x - 2i) = 0: x has 2 distinct complex roots.
Another example is the use of distinct is prime factors. The number 12 has a prime factorization of 2^{2}·3. Its distinct prime factors are 2 and 3.
To show that two objects are distinct, it helps to find some property of the two numbers that is different. For example, if x has two roots, 2 and 3, one could note that one of the values is even and the other is odd. This may seem silly in its simpleness, but in advanced math it is often useful.
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