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Solve for x: 3x⁴ - 8x² + 4 = 0. Which set gives all real roots?

1. x = +/-sqrt(2), +/-sqrt(2/3)
2. x = +/-2, +/-2/3
3. x = +/-sqrt(2), +/-1/3
4. x = +/-1, +/-2/3

Correct answer: x = +/-sqrt(2), +/-sqrt(2/3)

Solution

Substituting y = x² turns the equation into 3y² - 8y + 4 = 0, which factors to (3y - 2)(y - 2) = 0, giving y = 2/3 or y = 2. Taking square roots gives the four real roots x = +/-sqrt(2) and x = +/-sqrt(2/3).

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