Solve each equation in Exercises 83–108 by the method of your choice. x^2 - 6x + 13 = 0

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 these equations 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 effectively.

Completing the Square

Completing the square is a method used to solve quadratic equations by rewriting the equation in the form (x - p)^2 = q. This technique involves manipulating the equation to create a perfect square trinomial, which makes it easier to solve for x. It is particularly useful when the quadratic does not factor easily.

Discriminant

The discriminant of a quadratic equation, given by the formula D = b^2 - 4ac, determines the nature of the roots of the equation. If D > 0, there are two distinct real roots; if D = 0, there is one real root (a repeated root); and if D < 0, the roots are complex and not real. Analyzing the discriminant helps in understanding the solutions' characteristics.