Answer each question. Find the values of a, b, and c for which the quadratic equation. ax^2 + bx + c = 0 has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.) i, -i

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to this equation, known as roots, can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for solving them.

Zero-Product Property

The zero-product property states that if the product of two factors equals zero, then at least one of the factors must be zero. This principle is crucial when solving quadratic equations, as it allows us to set each factor equal to zero to find the solutions. In this context, it helps in determining the coefficients a, b, and c when given the roots.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'i' is the imaginary unit defined as the square root of -1. In this problem, the solutions i and -i are complex conjugates, which imply that the quadratic can be factored into real coefficients. Understanding complex numbers is essential for working with equations that have non-real solutions.