All of the equations we have solved so far have had...

All of the equations we have solved so far have had rational-number coefficients. However, the quadratic formula can be used to solve quadratic equations with irrational or even imaginary coefficients. Solve each equation.
$$
100 i x^{2}+300 x-200 i=0
$$

Key Concepts

Discriminant

The discriminant, given by b² - 4ac in the quadratic formula, indicates the nature of the roots. When working with quadratic equations that have complex or irrational coefficients, the discriminant may be negative or complex, indicating non-real or complex conjugate roots. Analyzing the discriminant helps determine the types of solutions and informs the subsequent steps in simplifying the formula.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically written in the form ax² + bx + c = 0. They appear in various contexts, and the methods for solving them—including factoring, completing the square, and using the quadratic formula—apply universally, even when the coefficients are irrational or complex.

Quadratic Formula

The quadratic formula, x = (-b ± ?(b² - 4ac)) / (2a), is a widely used method to solve any quadratic equation. It provides a systematic approach that works regardless of whether the coefficients are rational, irrational, or complex, thus ensuring a solution exists as long as the calculations are correctly managed.

Complex Numbers

Complex numbers extend the real numbers by including the imaginary unit i, where i² = -1. They are essential in solving equations with non-real coefficients or solutions. Understanding the arithmetic of complex numbers, including addition, multiplication, and taking square roots, is crucial for applying the quadratic formula when complex numbers are involved.

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