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Step 1:I'll solve this step by step, explaining why the solutions to x^{2} = 17 are irrational.
Step 2:: Identify the equation
The equation is $$x^{2} = 17$$, which means we need to find the values of $$x$$ that, when squared, equal 17.
Step 3:: Solve for x
x = \pm\sqrt{17}
Step 4:: Analyze the nature of \sqrt{17}
To prove $$\sqrt{17}$$ is irrational, we'll use a proof by contradiction.
Step 5:: Proof by Contradiction
- Rearrange: $$17b^{2} = a^{2}
- This means \sqrt{17} = \frac{a}{b}, where a and b are integers with no common factors
Step 6:: Contradiction
- This would imply $$a^{2}$$ is divisible by 17
- But this is impossible because 17 is a prime number - Therefore, our initial assumption must be false
Final Answer
The solutions \pm\sqrt{17} are irrational because \sqrt{17} cannot be expressed as a ratio of integers.
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