If b²-4ac < 0
There are no roots
When you have b²-4ac < 0 (where b, a, and c are constants), you have what is called a complex quadratic. In other words, the quadratic equation has no real solutions. To see why, recall the quadratic formula: x = (-b ± √b²-4ac) / 2a The term under the square root, b²-4ac, is called the discriminant. If the discriminant is negative, then taking the square root will involve imaginary numbers, represented by the imaginary unit i (where i² = -1). So, if b²-4ac < 0, then we have: √(b²-4ac) = √(-1) * √(-(b²-4ac)) = i * √(-b²+4ac) Plugging this into the quadratic formula, we get: x = (-b ± i * √(-b²+4ac)) / 2a This is the general form of the solution to a complex quadratic equation. You could simplify the expression further by dividing the numerator and denominator by 2, but the basic form remains the same. Note that if you plot a graph of a complex quadratic, it will not cross the x-axis, as it has no real solutions. Instead, it will stay above or below the x-axis, depending on the sign of the leading coefficient (a).
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