Quadratic Equation Solver (Quadratic Formula & pq-Formula)
Solve quadratic equations step by step with discriminant calculation, vertex coordinates, real/complex roots, and factored form.
Note: You can use both comma (,) and dot (.) as decimal separators.
Mathematical Explanation
Eine quadratische Gleichung ist eine algebraische Gleichung zweiten Grades der allgemeinen Form:
Die Mitternachtsformel (abc-Formel):
Zur universellen Lösung dient die abc-Formel:
Der Term unter der Wurzel, D = b² - 4ac, wird als Diskriminante bezeichnet:
- D > 0: Es gibt zwei verschiedene reelle Lösungen (zwei Nullstellen).
- D = 0: Es gibt genau eine reelle Lösung (eine doppelte Nullstelle / Berührpunkt).
- D < 0: Es gibt keine reellen Nullstellen (zwei konjugiert komplexe Lösungen).
Historical Origin & The Challenge Solved
Why was this mathematics developed, where does it come from, and what real-world problem did it solve?
1. Land Surveying, Field Allocation & Inheritance
Builders and administrators in ancient Babylon and Egypt faced the practical challenge of dividing rectangular plots with fixed area and perimeter constraints. Additionally, 9th-century Islamic inheritance law required exact quadratic calculations to distribute complex estates among heirs.
2. Geometric Completing the Square
In 820 CE, Muhammad ibn Musa al-Khwarizmi authored "Al-Kitab al-muhtasar fi hisab al-gabr wal-muqabala" (the origin of the word algebra). He systematically solved quadratic equations by geometrically completing squares, which eventually evolved into the quadratic formula.
3. Ballistics, Architecture & Physics
Every projectile trajectory, satellite dish design (parabolic reflectors), and suspension bridge cable precisely follows quadratic equations.
Real-World Applications & Practical Use Cases
Why do we learn this? Here is how this mathematics is applied in business, engineering, science, and everyday life:
Projectile Trajectories & Ballistics
Every thrown or propelled object (basketball, artillery shell, rocket) follows a parabolic path y = ax² + bx + c influenced by gravity.
The parabola vertex S(d | e) defines maximum flight altitude, while positive roots pinpoint exact landing locations.
Satellite Dishes & Suspension Bridges
Parabolic reflectors focus incoming parallel satellite signals directly onto the focal receiver. Suspension bridge main cables hang in parabolas.
Satellite dishes utilize parabolic geometry to focus weak radio signals from orbit directly onto the LNB antenna head.
Price Optimization & Maximum Revenue
Revenue curves are quadratic: raising prices reduces consumer demand. The vertex pinpoint the optimum price that maximizes total revenue.
A software provider finds via parabolic modeling that a $19.90 monthly subscription yields peak monthly revenue.
Frequently Asked Questions (FAQ)
What does the discriminant D indicate?
If D > 0, there are two distinct real solutions. If D = 0, there is exactly one repeated real solution. If D < 0, there are two complex conjugate solutions.
What is the vertex of a parabola?
The vertex is the extreme point (minimum or maximum) of the parabola, located at x = -b / (2a).