Algebraic Expression Solver Calculator

Solve linear and quadratic algebraic expressions of the form ax² + bx + c = 0 or ax + b = 0. Get real solutions, step-by-step breakdowns, and visual graphs of your equation.

How to Use This Calculator

Enter coefficients for your algebraic expression (ax² + bx + c = 0)
Set coefficient 'a' to 0 if solving a linear equation (bx + c = 0)
Check "Show step-by-step solution" for detailed working
Click "Solve Equation" to get roots, discriminant, and graph

Formula Used

x = (-b ± √(b² - 4ac)) / (2a)

Where:

a = Coefficient of x² term
b = Coefficient of x term
c = Constant term
Discriminant (D) = b² - 4ac (determines nature of roots)

Example Calculation

Real-World Scenario:

A projectile's height follows the equation h(t) = -5t² + 20t + 15, where h is height in meters and t is time in seconds. When does it hit the ground (h = 0)?

Given:

a = -5 (coefficient of t²)
b = 20 (coefficient of t)
c = 15 (constant term)

Calculation:

Discriminant D = 20² - 4(-5)(15) = 400 + 300 = 700

t = [-20 ± √700] / [2(-5)] = [-20 ± 26.46] / [-10]

Result: t = 4.65 seconds (positive root only, since time can't be negative)

Why This Calculation Matters

Practical Applications

Physics: Calculating projectile motion and trajectories
Engineering: Designing parabolic structures and systems
Economics: Finding break-even points in profit models

Key Benefits

Instantly solves complex equations that are tedious by hand
Visualizes the equation with an interactive graph
Provides educational step-by-step solutions for learning

Common Mistakes & Tips

Always ensure your equation is in standard form (ax² + bx + c = 0). If a term is missing (like no x term), set its coefficient to 0, not leave it blank. For example, x² + 4 = 0 should be entered as a=1, b=0, c=4.

When the discriminant (b² - 4ac) is negative, the equation has no real solutions - only complex ones. This doesn't mean your calculation is wrong; it means the parabola never crosses the x-axis. The calculator will clearly indicate when solutions are complex.

Frequently Asked Questions

No, this calculator is specifically designed for linear (degree 1) and quadratic (degree 2) equations. For higher-degree polynomials, you would need a different solver that uses numerical methods or factorization techniques.

When the discriminant equals zero (b² - 4ac = 0), there is exactly one real solution, called a repeated or double root. Graphically, this means the parabola touches the x-axis at exactly one point (the vertex).

Solutions are calculated with high precision (typically 10+ decimal places internally) and displayed with appropriate rounding. For exact values (like √2), the calculator shows both exact forms when possible and decimal approximations for practical use.

References & Disclaimer

Educational Disclaimer

This calculator is intended for educational and illustrative purposes only. While every effort has been made to ensure accuracy, users should verify critical calculations independently, especially for academic submissions or real-world applications where precision is essential.

References

Khan Academy: Quadratic Equations - Comprehensive tutorials on solving quadratic equations
Wolfram MathWorld: Quadratic Equation - Mathematical reference with historical context and advanced concepts
Purplemath: The Quadratic Formula - Step-by-step explanations with practical examples

Accuracy Notice

This calculator uses the standard quadratic formula and handles floating-point arithmetic with standard JavaScript precision. Extremely large or small coefficients may lead to rounding errors. Complex solutions are displayed in standard a + bi form. The graph displays real-valued functions only and may not show complex behavior.

About the Author

Kumaravel Madhavan

Web developer and data researcher creating accurate, easy-to-use calculators across health, finance, education, and construction and more. Works with subject-matter experts to ensure formulas meet trusted standards like WHO, NIH, and ISO.

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