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If the quadratic equation bx² + cx + a = 0 has non-real roots, then for every real x, the quantity 3b²x² + 6bcx + 2c² is:

1. less than 4ab
2. greater than −4ab
3. less than −4ab
4. greater than 4ab

Correct answer: greater than −4ab

Solution

The quadratic equation has non-real roots when its discriminant is negative, which implies that the parabola opens upwards and does not intersect the x-axis. Therefore, the expression 3b²x² + 6bcx + 2c², being a quadratic in x with positive leading coefficient, will always yield values greater than a certain minimum, which in this case is greater than -4ab.

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