1.10
Quadratic Equations in the Complex Number System
A quadratic equation in the form ax2+bx+c=0 can have solutions that vary in nature depending on the value of the discriminant, b2−4ac. In this express…
Quadratic equations in which the discriminant —the expression under the square root in the quadratic formula—has a negative value have no real solutions.
To solve these, the number system is extended to include complex numbers using the imaginary unit i, defined as the square root of a negative one.
The discriminant determines the solution's nature; a positive discriminant gives two unequal real solutions.
A zero discriminant gives one repeated real solution, meaning both solutions from the quadratic formula are equal.
A negative discriminant gives two complex solutions: non-real and complex conjugates—meaning, complex number pairs with equal real parts and imaginary parts with opposite sign.
An example of such a quadratic equation has solutions including the imaginary unit i, resulting from a negative discriminant.
Substituting these complex solutions into the original equation gives same values on both sides, confirming their validity.
Consider a rectangle with a perimeter of 20 cm, area of 30 cm². Substituting the perimeter equation into the area expression gives a quadratic equation with a negative discriminant, leading to complex roots—showing no real rectangle can exist.
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Q1: What does the discriminant tell you about a quadratic equation's solutions?
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.
Q2: How do you solve quadratic equations with negative discriminants?
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.
Q3: What are complex conjugates in quadratic solutions?
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.
Q4: Can a real rectangle have dimensions satisfying both a 20 cm perimeter and 30 cm² area?
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.
Q5: How do you verify that a complex number is a valid solution to a quadratic equation?
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.
Q6: Why is the imaginary unit i necessary for solving certain quadratic equations?
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.
Q7: What is the relationship between the quadratic formula and the discriminant?
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.