Quadratic Equations: Positive Discriminant And Its Significance

If b²-4ac < 0

There are no roots

If b²-4ac is less than zero, it means that the discriminant is negative. This discriminant is a value used to determine the nature of the roots of a quadratic equation. When the discriminant is negative, it implies that the equation has no real roots, but two complex conjugate roots.

In other words, the quadratic equation in standard form, ax²+bx+c=0, where a, b, and c are real numbers, has roots that are complex numbers of the form (p+qi) and (p-qi), where p and q are real numbers and i is the imaginary unit, such that i²=-1.

For example, consider the equation x²+2x+2=0. Here a=1, b=2, and c=2. Using the quadratic formula:

x = (-2 ± sqrt(2²-4(1)(2))) / (2(1))
x = (-2 ± sqrt(-4)) / 2
x = (-2 ± 2i) / 2

Thus the roots of the quadratic equation are (-1 + i) and (-1 – i), which are complex conjugates.

Therefore, if b²-4ac is less than zero, the quadratic equation has two complex roots that cannot be expressed as real numbers.

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