Solve 3x^2 − 12x + 12 ≤ 0.

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Multiple Choice

Solve 3x^2 − 12x + 12 ≤ 0.

Explanation:
Treating the left side as a perfect square makes the idea clear: a square is always nonnegative, so to be less than or equal to zero it must be exactly zero. Factor the quadratic: 3x^2 − 12x + 12 = 3(x^2 − 4x + 4) = 3(x − 2)^2. The inequality becomes 3(x − 2)^2 ≤ 0. Since (x − 2)^2 ≥ 0 for all x, the only way this is ≤ 0 is when (x − 2)^2 = 0, which gives x = 2. So the value that works is 2.

Treating the left side as a perfect square makes the idea clear: a square is always nonnegative, so to be less than or equal to zero it must be exactly zero. Factor the quadratic: 3x^2 − 12x + 12 = 3(x^2 − 4x + 4) = 3(x − 2)^2. The inequality becomes 3(x − 2)^2 ≤ 0. Since (x − 2)^2 ≥ 0 for all x, the only way this is ≤ 0 is when (x − 2)^2 = 0, which gives x = 2. So the value that works is 2.

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