{
  "problem_id": "math_1",
  "domain": "math",
  "difficulty": "hard",
  "prompt": "Find all real solutions to the equation x^4 - 5x^2 + 6 = 0. Show your complete reasoning.",
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  "thinking_text": "\nOkay, so I need to solve this equation: x\u2074 - 5x\u00b2 + 6 = 0. Hmm, let me think. It looks like a quartic equation, but maybe I can simplify it somehow. Let me see... The equation has x\u2074, x\u00b2, and a constant term. That makes me think it's a quadratic in terms of x\u00b2. Maybe I can use substitution here. Let me try letting y = x\u00b2. Then, the equation would become y\u00b2 - 5y + 6 = 0. Oh, that seems manageable! \n\nAlright, so if I have y\u00b2 - 5y + 6 = 0, I can try to factor this quadratic. Let me check if it factors nicely. Looking for two numbers that multiply to 6 and add up to -5. Let me think... The factors of 6 are 1 and 6, 2 and 3. Since the middle term is -5y, maybe both numbers are negative? Let me check: -2 and -3 multiply to +6 and add up to -5. Yes! So, the quadratic factors as (y - 2)(y - 3) = 0. Wait, no, if the numbers are -2 and -3, then the factors would be (y - 2)(y - 3)? Wait, hold on. Let me verify that. If I expand (y - 2)(y - 3), I get y\u00b2 - 3y - 2y + 6 = y\u00b2 - 5y + 6. Yes, that's correct. So, the solutions for y would be y = 2 and y = 3. \n\nBut remember that y = x\u00b2, so substituting back, we have x\u00b2 = 2 and x\u00b2 = 3. To find the real solutions for x, I need to solve these equations. For x\u00b2 = 2, taking the square root of both sides gives x = \u00b1\u221a2. Similarly, for x\u00b2 = 3, taking the square root gives x = \u00b1\u221a3. \n\nSo, putting it all together, the real solutions to the original equation are x = \u221a2, x = -\u221a2, x = \u221a3, and x = -\u221a3. Let me check if these solutions actually work in the original equation to make sure I didn't make any mistakes. \n\nFirst, let's check x = \u221a2. Plugging into x\u2074 - 5x\u00b2 + 6: (\u221a2)^4 - 5*(\u221a2)^2 + 6. (\u221a2)^4 is ((\u221a2)^2)^2 = (2)^2 = 4. Then 5*(\u221a2)^2 is 5*2 = 10. So the expression becomes 4 - 10 + 6 = 0. Perfect, that works. \n\nNow x = -\u221a2. Since even powers will make it positive, (-\u221a2)^4 is also 4, and (-\u221a2)^2 is 2, so same calculation: 4 - 10 + 6 = 0. Good. \n\nNext, x = \u221a3. (\u221a3)^4 is ((\u221a3)^2)^2 = 3\u00b2 = 9. Then 5*(\u221a3)^2 is 5*3 = 15. So 9 - 15 + 6 = 0. Correct. \n\nSi",
  "response_text": "\n\nTo solve the equation:\n\n$$\nx^4 - 5x^2 + 6 = 0\n$$\n\nwe begin by recognizing that it is a **biquadratic equation**\u2014a quartic equation that involves only even powers of $ x $. This allows us to simplify the equation by using a substitution.\n\n---\n\n### Step 1: Substitution\n\nLet:\n\n$$\ny = x^2\n$$\n\nThen the original equation becomes:\n\n$$\ny^2 - 5y + 6 = 0\n$$\n\n---\n\n### Step 2: Solve the Quadratic\n\nWe solve the quadratic equation:\n\n$$\ny^2 - 5y + 6 = 0\n$$\n\nFactoring:\n\n$$\n(y - 2)(y - 3) = 0\n$$\n\nSo the solutions are:\n\n$$\ny = 2 \\quad \\text{or} \\quad y = 3\n$$\n\n---\n\n### Step 3: Back-Substitute and Solve for $ x $\n\nRecall that $ y = x^2 $, so:\n\n1. If $ x^2 = 2 $, then $ x = \\pm \\sqrt{2} $\n2. If $ x^2 = 3 $, then $ x = \\pm \\sqrt{3} $\n\n---\n\n### Step 4: Verify the Solutions\n\nLet\u2019s verify one of the roots in the original equation to be sure:\n\nTake $ x = \\sqrt{2} $:\n\n$$\nx^4 - 5x^2 + 6 = (\\sqrt{2})^4 - 5(\\sqrt{2})^2 + 6 = 4 - 10 + 6 = 0\n$$\n\nTrue. Similarly, all other values of $ x $ will yield the same result due to the symmetry and even powers involved.\n\n---\n\n### Final Answer\n\nThe real solutions to the equation $ x^4 - 5x^2 + 6 = 0 $ are:\n\n$$\n\\boxed{-\\sqrt{3}}, \\quad \\boxed{-\\sqrt{2}}, \\quad \\boxed{\\sqrt{2}}, \\quad \\boxed{\\sqrt{3}}\n$$<|im_end|>",
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