Quadratic Equations
Introduction to quadratic functions
Start with the basics and progress through 7 lessons. Each lesson builds on the previous one.
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Test Your Knowledge
10 questions, new mix each time (from 108)
In This Topic (7 lessons)
Solving Quadratic Equations by Square Roots
Learn to solve quadratic equations by taking the square root of both sides.
Introduction to Quadratic Equations
Learn what quadratic equations are, their standard form, and how to identify their key features.
Vertex Form
Learn to write and interpret quadratic functions in vertex form to easily identify the vertex and graph parabolas.
Completing the Square
Master the technique of completing the square to solve quadratic equations and rewrite them in vertex form.
The Quadratic Formula
Learn to solve any quadratic equation using the quadratic formula.
The Discriminant
Learn how the discriminant reveals the nature of quadratic solutions before solving.
Quadratic Word Problems
Learn to translate real-world situations into quadratic equations and solve them to find meaningful answers.
Quadratic functions and equations are central to algebra and appear throughout mathematics and science. A quadratic equation has the form ax² + bx + c = 0, and its graph is a parabola—a U-shaped curve that models everything from thrown balls to satellite dishes. Understanding quadratics is essential for physics, engineering, and advanced mathematics.
In this topic, you will master multiple methods for solving quadratic equations: factoring, completing the square, and the quadratic formula. You will learn when each method is most efficient. Graphing parabolas, finding vertices, and understanding how coefficients affect the shape all deepen your understanding of these fundamental functions.
Our lessons connect quadratics to real applications: projectile motion, optimization problems, and area calculations. The discriminant reveals the nature of solutions before you solve. With practice, you will develop both computational skills and conceptual insight into quadratic behavior.
What You'll Learn
- Solve quadratic equations by factoring
- Apply the quadratic formula for any quadratic equation
- Complete the square to solve equations and vertex form
- Graph parabolas by identifying vertex and intercepts
- Use the discriminant to determine the nature of solutions
- Solve quadratic word problems involving area and projectiles
- Convert between standard, vertex, and factored forms
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula x = (-b ± √(b²-4ac)) / 2a gives solutions to ax² + bx + c = 0. It works for any quadratic equation, even when factoring is difficult or impossible.
What does the discriminant tell you?
The discriminant b² - 4ac reveals the nature of solutions. If positive, there are two real solutions. If zero, one real solution (a repeated root). If negative, no real solutions (two complex solutions).
When should I complete the square vs. use the formula?
Complete the square when converting to vertex form or when the equation is close to a perfect square. Use the formula when coefficients are complicated or when you just need quick solutions.