Quadratic Equation Solver
Solve any quadratic equation with step-by-step solutions
Find roots, discriminant, vertex, and visualize parabolas with detailed analysis
Select formula and enter parameters
Formula form
Results
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📖 How to use this calculator
Enter coefficients
Input the values of A, B, and C from your quadratic equation in the form Ax² + Bx + C = 0. Remember that A cannot be zero.
View results
The calculator automatically computes the discriminant and all solutions (real or complex) with detailed mathematical steps.
Visualize the parabola
See a graphical representation of your quadratic function with the vertex and x-intercepts clearly marked.
❓ What is a quadratic equation?
A quadratic equation is a polynomial equation of the second degree. It has the general form:
Ax² + Bx + C = 0
Where:
- A is the coefficient of x² (A ≠ 0)
- B is the coefficient of x
- C is the constant term
Quadratic equations appear frequently in mathematics, physics, engineering, and many real-world applications like projectile motion, optimization problems, and circuit analysis.
🔄 Three forms of quadratic equations
Quadratic equations can be written in three equivalent forms. This calculator supports all three:
📐 Standard Form
Ax² + Bx + C = 0
This is the most common form. It's useful for directly applying the quadratic formula and calculating the discriminant.
📍 Vertex Form
A(x - H)² + K = 0
This form reveals the vertex (H, K) of the parabola directly. It's ideal for graphing and understanding the parabola's transformation.
✂️ Factored Form
A(x - x₁)(x - x₂) = 0
This form shows the solutions (roots) x₁ and x₂ directly. For complex roots, x₁ represents the real part and x₂ the imaginary coefficient (when "Allow negative discriminant" is enabled).
💡 Pro Tip: You can switch between these forms at any time using the dropdown menu. The calculator automatically converts your values to the selected form!
📝 How to solve a quadratic equation
The quadratic formula is used to find the solutions (roots) of any quadratic equation:
x = (-B ± √Δ) / (2A)
where Δ = B² - 4AC
📚 Example: Solve 2x² + 5x - 3 = 0
Step 1: Identify the coefficients
A = 2, B = 5, C = -3
Step 2: Calculate the discriminant
Δ = B² - 4AC
Δ = (5)² - 4(2)(-3)
Δ = 25 + 24
Δ = 49
Step 3: Apply the quadratic formula
x = (-B ± √Δ) / (2A)
x = (-5 ± √49) / (2 × 2)
x = (-5 ± 7) / 4
Step 4: Find both solutions
x₁ = (-5 + 7) / 4 = 2 / 4 = 0.5
x₂ = (-5 - 7) / 4 = -12 / 4 = -3
Step 5: Verify the solutions
For x = 0.5: 2(0.5)² + 5(0.5) - 3 = 0.5 + 2.5 - 3 = 0 ✓
For x = -3: 2(-3)² + 5(-3) - 3 = 18 - 15 - 3 = 0 ✓
💡 Tip: Use the "Show step by step solution" checkbox above to see detailed calculations for any equation you enter!
🔑 Understanding the discriminant
The discriminant (Δ) is a value that determines the nature of the solutions of a quadratic equation:
Δ = B² - 4AC
If Δ > 0:
The equation has two distinct real solutions.
If Δ = 0:
The equation has one repeated real solution (the parabola touches the x-axis at one point).
If Δ < 0:
The equation has two complex conjugate solutions (the parabola doesn't intersect the x-axis).
🎯Interesting Discovery: Discriminant in Vertex Form
When using the vertex form A(x - H)² + K, the discriminant simplifies to a beautiful formula:
Δ = -4AK
Key insight: The discriminant depends only on the coefficient A and the vertex's y-coordinate K, but is completely independent of the vertex's x-coordinate H!
This means sliding the parabola left or right (changing H) doesn't affect whether it crosses the x-axis, only moving it up or down (changing K) does. Try it yourself in vertex form!
❔ FAQs
Q: What does it mean when A = 0?
A: If A = 0, the equation is no longer quadratic but linear. This calculator requires A ≠ 0 to solve quadratic equations.
Q: Can I use this calculator for negative coefficients?
A: Yes! The calculator fully supports negative coefficients. Simply enter them with a minus sign (e.g., -5, -3).
Q: What are complex solutions?
A: Complex solutions contain imaginary numbers (involving i, where i² = -1). They occur when the discriminant is negative. Enable "Allow negative discriminant" to see them.
Q: What is the vertex of a parabola?
A: The vertex is the highest or lowest point on the parabola. It's where x = -B/(2A) and provides important information about the function's minimum or maximum value.
Q: Can I enter complex roots in factored form A(x - x₁)(x - x₂) = 0?
A: Yes! When you switch to factored form from an equation with complex solutions, the input fields automatically change to accept complex conjugate pairs. You'll enter the real part (a) and imaginary coefficient (b) to represent roots a ± bi. For real roots, the fields display as x₁ and x₂ normally.
Q: Are there any limits on the input values?
A: Yes, for optimal performance and numerical stability, all input values must be between -1,000,000 and 1,000,000. Additionally, the coefficient A must have an absolute value of at least 0.000001 to prevent numerical instability. These limits ensure accurate calculations and prevent browser performance issues.
Q: How can I verify my answer?
A: Substitute your solution back into the original equation. If it equals zero, your answer is correct. The step-by-step solution can help you verify each calculation.