Answer:
The solution to the equation x^2=2 is an irrational number. Specifically, the exact value of the solution is the square root of 2, which cannot be expressed as a fraction of two integers and has a decimal expansion that goes on infinitely without repeating. Therefore, the correct answer is D.
Explanation:
The equation x^2=2 can be solved by taking the square root of both sides of the equation. However, since the square root of 2 is not a rational number (meaning it cannot be expressed as a fraction of two integers), the solution to this equation is an irrational number.
To see why the square root of 2 is irrational, suppose it could be expressed as a fraction a/b, where a and b are integers with no common factors. Then, squaring both sides of this equation gives:
(a/b)^2 = 2
a^2/b^2 = 2
Multiplying both sides by b^2 gives:
a^2 = 2b^2
This means that a^2 is an even number, since it is equal to twice an integer (namely, 2b^2). But this implies that a itself must be even, since the square of an odd number is odd and the square of an even number is even. Therefore, we can write a=2k for some integer k.
Substituting this expression for a into the equation a^2=2b^2 gives:
(2k)^2 = 2b^2
4k^2 = 2b^2
2k^2 = b^2
This implies that b^2 is even, which means that b itself must be even (using the same argument as before). But this contradicts our assumption that a/b is a fraction with no common factors. Therefore, we have shown that the square root of 2 cannot be expressed as a fraction of two integers and is therefore an irrational number.
hope it helps!