Quadratic Equation Calculator
Quadratic Equation Calculator
Solve quadratic equations of the form ax² + bx + c = 0
Cannot be zero
Linear term
Constant term
Current Equation:
x² - 5x + 6 = 0
Quadratic Equation Calculator: Complete Guide
A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0.Quadratic equations are fundamental in algebra and appear throughout mathematics, physics, engineering, and many real-world applications. They describe parabolic relationships and are essential for optimization problems, projectile motion, and curve fitting.
Quick Answer
To solve a quadratic equation: Enter coefficients a, b, and c for ax² + bx + c = 0. The calculator uses the quadratic formula x = (-b ± √(b²-4ac))/2a to find roots, analyzes the discriminant to determine the nature of solutions, and provides vertex form and graphing information.
Mathematical Foundation
The quadratic formula - the universal method for solving ax² + bx + c = 0
Key Concepts:
Standard Form
ax² + bx + c = 0 where a, b, c are constants and a ≠ 0. The coefficient 'a' determines the parabola's direction (upward if a > 0, downward if a < 0) and width.
Discriminant
Δ = b² - 4ac determines the nature of roots: Δ > 0 (two real roots), Δ = 0 (one repeated root), Δ < 0 (two complex roots). The discriminant also indicates if the parabola intersects the x-axis.
Vertex Form
y = a(x - h)² + k where (h,k) is the vertex. This form clearly shows the parabola's turning point and makes it easy to identify the axis of symmetry x = h and the maximum or minimum value k.
Methods for Solving Quadratic Equations
Quadratic Formula (Universal Method)
Works for all quadratic equations, regardless of factorability.
x = (-b ± √(b²-4ac)) / 2aFactoring Method
Express as (px + q)(rx + s) = 0, then use zero product property.
ax² + bx + c = (mx + n)(px + q) = 0Completing the Square
Transform to (x + d)² = e form, then take square roots.
x² + bx = -(c/a) → (x + b/2)² = (b/2)² - c/aApplications of Quadratic Equations
Physics & Engineering
Projectile Motion
Height equation: h(t) = -16t² + v₀t + h₀ for objects under gravity
Optimization Problems
Maximize/minimize area, volume, profit using parabolic relationships
Electrical Circuits
AC circuit analysis, resonance frequency calculations
Structural Design
Beam deflection, arch design, suspension bridge calculations
Business & Economics
Profit Maximization
Revenue R = px, Cost C = ax² + bx + c, find optimal production level
Break-even Analysis
Find production levels where total revenue equals total cost
Supply and Demand
Model market equilibrium with quadratic supply/demand curves
Investment Growth
Compound interest models, portfolio optimization problems
Example Problems with Solutions
Example 1: Solving by Quadratic Formula
Solve: 2x² - 7x + 3 = 0
Answer: x = 3 or x = 1/2
Example 2: Factoring Method
Solve: x² - 5x + 6 = 0
Answer: x = 2 or x = 3
Example 3: Projectile Motion Application
A ball is thrown upward with initial velocity 64 ft/s from height 6 ft. When does it hit the ground? (Use h(t) = -16t² + 64t + 6)
Answer: The ball hits the ground after approximately 4.09 seconds
Understanding the Discriminant
Δ > 0: Two Real Roots
The parabola crosses the x-axis at two distinct points.
x₂ = (-b - √Δ) / 2a
Δ = 9 - 8 = 1 > 0
Δ = 0: One Real Root
The parabola touches the x-axis at exactly one point (vertex).
(repeated root)
Δ = 16 - 16 = 0
Δ < 0: No Real Roots
The parabola doesn't intersect the x-axis. Roots are complex.
(complex conjugates)
Δ = 1 - 4 = -3 < 0
Vertex Form and Graph Analysis
Finding the Vertex
The vertex (h,k) is the turning point of the parabola.
Graph Characteristics
Calculator Input Guide
Coefficient Entry
Common Formats
Important Notes
- • The coefficient 'a' must be non-zero (otherwise it's not quadratic)
- • Use negative signs for coefficients, not within the input fields
- • Decimal coefficients are fully supported for precision
- • The calculator automatically determines the best solution method
- • Complex roots are displayed when discriminant is negative
Common Mistakes and Solutions
Common Errors
- Sign errors: Incorrectly handling negative coefficients
- Discriminant calculation: Arithmetic mistakes in b²-4ac
- Order of operations: Incorrect precedence in formula
- Incomplete solutions: Finding only one root when two exist
Best Practices
- Double-check coefficient signs and values
- Verify discriminant calculation step by step
- Check answers by substituting back into original equation
- Use multiple solution methods to verify results
Frequently Asked Questions
What makes an equation quadratic?
A quadratic equation must have the form ax² + bx + c = 0 where 'a' is not zero. The highest power of the variable is 2 (squared term). If a = 0, the equation becomes linear (bx + c = 0). The presence of the x² term is what makes it quadratic, regardless of whether the other terms are present.
Why does the quadratic formula always work?
The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0. This algebraic manipulation is always valid (when a ≠ 0), making the formula universal. Other methods like factoring only work when the equation has nice integer factors, but the quadratic formula handles all cases, including those with irrational or complex roots.
What does it mean when the discriminant is negative?
When Δ = b² - 4ac < 0, the quadratic equation has no real solutions. Instead, it has two complex conjugate roots of the form x = p ± qi, where i = √(-1). Geometrically, this means the parabola doesn't cross the x-axis. The complex roots are still mathematically valid and important in advanced applications.
How is the vertex related to the roots?
The vertex x-coordinate h = -b/2a is always exactly halfway between the two roots (when they exist). If the roots are x₁ and x₂, then h = (x₁ + x₂)/2. The vertex represents the maximum or minimum value of the quadratic function, and its x-coordinate is the axis of symmetry of the parabola.
When should I use factoring vs. the quadratic formula?
Use factoring when the equation has obvious integer factors or when coefficients are small integers. Use the quadratic formula when factoring isn't obvious, when dealing with decimal coefficients, or when you need exact answers. For learning purposes, try factoring first, but use the formula as a reliable backup. The formula is also preferred for verifying factored solutions.
Can a quadratic equation have more than two solutions?
No, a quadratic equation can have at most two solutions. This is because the degree of the polynomial (highest power) determines the maximum number of roots. A quadratic (degree 2) has exactly 2 roots when Δ > 0, exactly 1 repeated root when Δ = 0, and 0 real roots (but 2 complex roots) when Δ < 0. This is guaranteed by the Fundamental Theorem of Algebra.
How accurate are decimal approximations of roots?
The calculator provides high precision decimal approximations typically accurate to 10+ decimal places. For exact results involving square roots (like √5 or √13), the calculator shows both the exact form and decimal approximation. When discriminants are perfect squares, the roots are exactly rational and no approximation is needed. For most practical applications, the decimal precision is more than sufficient.
Advanced Topics in Quadratic Equations
Vieta's Formulas
Relationship between coefficients and roots without solving the equation:
Useful for checking solutions and constructing equations from known roots.
Nature of Roots Analysis
Beyond just counting roots, analyze their properties:
Quadratic Inequalities
Solving ax² + bx + c > 0 or ax² + bx + c < 0:
Related Mathematical Tools
Quadratic Equation Calculator
Solve quadratic equations of the form ax² + bx + c = 0
Cannot be zero
Linear term
Constant term
Current Equation:
x² - 5x + 6 = 0