Solve each equation by completing the square. Write complex...

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in one variable, typically expressed in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ? 0. Understanding the characteristics of quadratic equations is essential for selecting and applying various solution techniques.

Standard Form

Writing a quadratic equation in standard form (ax² + bx + c = 0) organizes the terms in a consistent manner, making it easier to apply solution methods such as completing the square or the quadratic formula.

Completing the Square

Completing the square is a method used to solve quadratic equations by converting the quadratic expression into a perfect square trinomial plus or minus a constant. This technique simplifies the process of isolating the variable and is especially useful for deriving the vertex form of a parabola.

Complex Numbers

Complex numbers consist of a real part and an imaginary part, typically written in the form a ± b i, where i is the imaginary unit satisfying i² = -1. They are crucial for finding solutions to quadratic equations when the discriminant is negative, leading to non-real roots.

Discriminant

The discriminant, given by b² – 4ac in a quadratic equation, determines the nature of the roots. A positive discriminant results in two distinct real roots, zero yields a repeated real root, and a negative discriminant indicates a pair of complex conjugate roots.

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