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In the equation 2x² − 4x = t, t is a constant. If the equation has no real solutions, then what is the minimum possible value of t?

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Final answer:

The minimum possible value of t for which the quadratic equation 2x² − 4x = t has no real solutions is any value greater than 2, as determined by the condition that the discriminant must be less than zero.

Step-by-step explanation:

To determine the minimum possible value of t in the equation 2x² − 4x = t where the equation has no real solutions, we must consider the discriminant of the quadratic equation. The discriminant (Δ) of a quadratic equation in the form of ax² + bx + c = 0 is b² - 4ac. For the equation to have no real solutions, the discriminant must be less than zero. Hence, we need (-4)² - 4(2)(t) < 0 which simplifies to 16 - 8t < 0.

Solving for t gives us t > 2. Therefore, the minimum possible value of t for which there are no real solutions is any value greater than 2. This is because at t = 2, the discriminant becomes zero and the quadratic equation has one real solution (a repeated root). To maintain a negative discriminant, corresponding to no real solutions, t must be strictly greater than 2.

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