Don’t panic when you see quadratic equations – they’re not as scary as they look! The standard form is ax² + bx + c = 0, and it models **any curved motion**: a basketball’s arc, a rocket’s path, or even a company’s profit curve.
By solving it, you can answer real questions like:
- When does the ball hit the ground?
- What’s the maximum height?
- What price gives the highest profit?
Try it now: Use our free Pythagorean Theorem calculator to test any equation instantly.
Method 1: Factoring (Fastest When It Works)
Example: Solve x² + 5x + 6 = 0
- Find two numbers that multiply to 6 and add to 5 → 2 and 3
- Rewrite:
(x + 2)(x + 3) = 0 - Set each factor to zero:
x = -2orx = -3
✅ Works only with nice numbers. Most equations need the formula.
Method 2: Quadratic Formula (Works Every Time)
Formula:
Example: 2x² + 4x - 6 = 0
- a = 2, b = 4, c = -6
- Discriminant: b² – 4ac = 16 + 48 = 64 → √64 = 8
- x = [-4 ± 8]/4 → x = 1 or x = -3
Discriminant tells you:
| Discriminant | Solutions |
|---|---|
| > 0 | 2 real roots |
| = 0 | 1 real root |
| < 0 | No real roots |
Method 3: Completing the Square (Best for Graphs)
Turns equation into vertex form: y = a(x - h)² + k
Vertex (h,k) = highest/lowest point → perfect for **optimization**.
Example: Max profit when price = x
- Profit = -2x² + 100x – 1000
- Vertex x = 25 → charge $25 for max profit
🚀 Real-World Examples (2025 Update)
- Physics: Basketball trajectory
- Engineering: Bridge arch design
- Business: Profit maximization
- Tech (2025): Drone delivery paths – companies use quadratic models to calculate perfect drop trajectory (10% efficiency gain in Q3 2025)
Google Doodle (Nov 2025): Interactive parabola tool went viral – millions learned quadratics in 5 minutes.
FAQ (Schema Included)
Why are quadratic equations important?
They model parabolas – the shape of motion under gravity, bridge arches, and profit curves. Used in physics, engineering, and business.
When to use formula vs factoring?
Factor only if numbers are simple. Quadratic formula works for all equations, especially with decimals or square roots.
What does the discriminant mean?
Tells number of real solutions: positive = 2, zero = 1, negative = none (parabola doesn’t cross x-axis).
Ready to Try?
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Pro Tip: Plug in real-world values – like initial velocity of a drone – and see the magic happen.