Quadratic Equations
Lessons in this chapter
About Quadratic Equations
Quadratic Equations teaches every standard method for solving ax² + bx + c = 0, each suited to different situations.
Properties of Radicals reviews simplifying square roots before Solving Quadratic Equations by Graphing finds solutions as the x-intercepts of the matching parabola, and Solving Quadratic Equations Using Square Roots handles the special case where a quadratic has no linear (x) term at all.
Solving Quadratic Equations by Completing the Square rewrites any quadratic into a perfect-square form to solve it directly, a method that's also used to derive Solving Quadratic Equations Using the Quadratic Formula's formula, x = (-b ± √(b² - 4ac)) / 2a, which solves any quadratic equation regardless of whether it factors nicely.
Solving Nonlinear Systems of Equations closes the chapter by finding where a quadratic and a linear equation (or two quadratics) intersect, combining system-solving skills with every quadratic-solving method from this chapter.
All six lessons in this chapter open as Algebra 1 worksheets in the page, including Solving Quadratic Equations by Completing the Square, Solving Quadratic Equations Using the Quadratic Formula and Solving Nonlinear Systems of Equations. Each question is marked as you answer it, which catches the most common slip in this chapter early: calculating only one of the two solutions and stopping. All six are free to try, with no login needed for your first 3.
Frequently asked questions
Properties of Radicals comes first as the simplifying skill the rest of the chapter needs. Solving Quadratic Equations by Graphing reads the solutions as the x-intercepts of a parabola, Solving Quadratic Equations Using Square Roots covers the special case with no linear term, and Solving Quadratic Equations by Completing the Square rewrites any quadratic into a perfect square. Solving Quadratic Equations Using the Quadratic Formula gives the method that works regardless of whether a quadratic factors, and Solving Nonlinear Systems of Equations finds where a parabola and a line meet. The chapter sits in the high school Algebra conceptual category (A).
Graphing (finding the x-intercepts), using square roots (when there’s no linear term), completing the square, and the quadratic formula: each suited to different situations.
When the equation has no linear (x) term, like x² - 16 = 0: isolate the squared term, then take the square root of both sides, remembering both the positive and negative roots.
It means adding a specific value to both sides of an equation so that one side becomes a perfect square trinomial, which can then be written as a squared binomial and solved using square roots.
The quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, solves any quadratic equation in standard form: it's the most universal method, useful especially when a quadratic doesn't factor nicely.
If b² - 4ac is positive, there are two real solutions; if it's zero, there's exactly one real solution; if it's negative, there are no real solutions (the graph never crosses the x-axis).