Quadratic Equations and Complex Numbers
Lessons in this chapter
About Quadratic Equations and Complex Numbers
Quadratic Equations and Complex Numbers covers every technique for solving quadratics, including the numbers needed when a solution isn’t real.
Solving Quadratic Equations opens with the standard solving techniques, before Complex Numbers introduces the imaginary unit needed when a quadratic has no real solutions.
Completing the Square and Using the Quadratic Formula cover two more general-purpose methods that solve any quadratic equation, real or complex solutions included.
Solving Nonlinear Systems and Quadratic Inequalities close the chapter by applying quadratics to systems involving a parabola and a line, and to inequalities where the solution is a range of values instead of a single number.
Each of the six lessons is an Algebra 2 worksheet worked in the page, including Completing the Square, Using the Quadratic Formula, and Quadratic Inequalities. The practice problems are scored as you answer them, which is useful here because most lost marks come from a sign slipping rather than from the method. Free to try, with no login needed for your first 3.
Frequently asked questions
Solving Quadratic Equations opens the chapter with factoring and square roots, and Complex Numbers then introduces i so that every quadratic has solutions. Completing the Square and Using the Quadratic Formula give the two methods that always work, Solving Nonlinear Systems applies them where a quadratic meets a line or another curve, and Quadratic Inequalities finishes with solution regions rather than single values. It sits in the Algebra (A) conceptual category of the high school Common Core, with complex numbers in Number and Quantity (N).
The imaginary unit i is defined as √-1, so i² = -1. It lets equations that have no real solution (like x² = -1) have a solution anyway.
A complex number has the form a + bi, combining a real part a and an imaginary part bi.
Completing the square rewrites a quadratic as a perfect square trinomial plus a constant, turning it into a form you can solve directly with a square root.
The quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, solves any quadratic equation ax² + bx + c = 0, whether its solutions are real or complex.
The discriminant, b² - 4ac, tells you the number and type of solutions without solving: positive means two real solutions, zero means one repeated real solution, and negative means two complex solutions.