Quadratic Equations
A quadratic equation has an x² term. Its graph is a parabola. Quadratics have up to two solutions (roots). They can be solved by factoring, completing the square, or the quadratic formula — one of the most important formulas in all of mathematics.
Key Concepts
Standard Form and Graphing
y = ax² + bx + c. When a > 0, the parabola opens up (has a minimum). When a < 0, it opens down (has a maximum). The vertex is the turning point. The x-intercepts are the solutions to ax² + bx + c = 0.
Solving by Factoring
x² + 5x + 6 = 0 → (x+2)(x+3) = 0. If a product is zero, one factor must be zero. x+2=0 → x=−2. x+3=0 → x=−3. Solutions: x = −2 or x = −3. Always set the equation equal to zero first.
The Quadratic Formula
x = (−b ± √(b²−4ac)) / 2a. Works for any quadratic. The discriminant b²−4ac tells you: if > 0, two real solutions; if = 0, one solution; if < 0, no real solutions. This formula is worth memorizing — it appears throughout math, physics, and engineering.
🆕 Quadratic Explorer
Adjust a, b, c and see the parabola, vertex, and solutions live!
✅ Check Your Understanding
1. The quadratic formula solves ax²+bx+c=0. What does the ± mean?
2. If the discriminant (b²−4ac) is negative, the equation has...
3. What shape does a quadratic function graph make?