Quadratic equations appear everywhere — from projectile motion to profit optimization. Any equation in the form ax² + bx + c = 0 (where a ≠ 0) is quadratic. This guide explains three solution methods and when to use each.
Standard Form and the Quadratic Formula
The quadratic formula x = (−b ± √(b² − 4ac)) / 2a solves any quadratic equation. The expression b² − 4ac is called the discriminant. If it is positive, you get two real roots. Zero gives one repeated root. Negative discriminant means complex roots.
Use our Quadratic Equation Calculator to compute roots instantly with step-by-step output.
Factoring Method
When the quadratic factors neatly over integers, factoring is fastest. For x² − 5x + 6 = 0, find two numbers that multiply to 6 and add to −5: −2 and −3. Thus (x − 2)(x − 3) = 0, giving x = 2 or x = 3.
Completing the Square
Completing the square transforms ax² + bx + c into vertex form a(x − h)² + k. This reveals the parabola's minimum or maximum — useful in optimization problems.
Real-World Applications
Quadratics model area problems, revenue curves, and ballistics. When height h(t) = −16t² + v₀t + h₀, finding when h = 0 tells you when a projectile lands.
Going Deeper
Learn how to solve quadratic equations using factoring, completing the square, and the quadratic formula. This guide connects theory to practice — use the related calculators linked at the bottom to verify each example with your own numbers.
Practical Tips
- Write down given values and unknowns before opening the calculator.
- Check units and rounding rules appropriate to your context (class, lab, or business).
- Compare manual working with the calculator result to build confidence.
Common Mistakes to Avoid
- Rushing inputs without reading field labels carefully.
- Confusing similar formulas that use different variables or units.
- Reporting results with more precision than your inputs justify.
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