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The discriminant, calculated as b²−4ac, determines the nature of a quadratic equation's solutions. A positive discriminant yields two unequal real solutions, a zero discriminant produces one repeated real solution, and a negative discriminant results in two complex conjugate solutions. This value guides whether solutions exist in the real number system or require complex numbers.

When a quadratic equation has a negative discriminant, the number system is extended using the imaginary unit i, defined as the square root of negative one. The square root of any negative number is rewritten using i; for example, √−36 becomes 6i. This allows solutions to be expressed as complex numbers with both real and imaginary components.

Complex conjugates are pairs of complex numbers with equal real parts and imaginary parts with opposite signs. When a quadratic equation has a negative discriminant, its two solutions form complex conjugate pairs. For instance, −3+2i and −3−2i are complex conjugates, both satisfying the original equation when substituted back.

No. Setting up equations for a rectangle with perimeter 20 cm and area 30 cm² produces a quadratic equation with a negative discriminant, yielding complex roots. Since dimensions must be real numbers, no actual rectangle can satisfy both constraints simultaneously, demonstrating how complex solutions reveal impossible real-world scenarios.

Substitute the complex number back into the original equation and simplify both sides. If both sides equal zero, the complex number is a valid solution. For example, substituting −3+2i into x²+6x+13=0 and expanding confirms the equation holds, verifying the solution's validity.

The imaginary unit i extends the number system beyond real numbers, enabling solutions to equations with negative discriminants. Without i, these equations would have no solutions. By defining i as √−1, mathematicians created complex numbers that allow all quadratic equations to have solutions, completing the mathematical framework.

The discriminant appears under the square root in the quadratic formula: x = (−b ± √(b²−4ac)) / 2a. Its value determines what type of answer emerges: real or complex. The discriminant's sign directly controls whether the square root yields a real number or requires the imaginary unit i to express the solution.