Equation Calculator | Quadratic & Linear Equation Solver

📖 How to Use This Equation Calculator

Select equation type - Choose Quadratic (ax² + bx + c = 0) or Linear (ax + b = 0).

Enter coefficients - Input the values for a, b, and c (quadratic) or a and b (linear).

Click "Solve" - Get the solution(s) with a detailed step-by-step explanation.

📐 Quadratic Equations (ax² + bx + c = 0)

A quadratic equation is a second-degree polynomial equation in a single variable. The standard form is ax² + bx + c = 0, where a ≠ 0. The solutions (roots) can be found using the quadratic formula.

🧮 Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

Discriminant (Δ = b² - 4ac):
• If Δ > 0: Two distinct real roots
• If Δ = 0: One real root (double root)
• If Δ < 0: Two complex conjugate roots

📈 Linear Equations (ax + b = 0)

A linear equation is a first-degree equation. The solution is simply x = -b/a (provided a ≠ 0). If a = 0 and b = 0, there are infinite solutions. If a = 0 and b ≠ 0, there is no solution.

💡 Tips for Solving Equations

Always check if a = 0 in quadratic equations (becomes linear)
The discriminant determines the nature of roots without solving completely
Complex roots always come in conjugate pairs (a ± bi)
For quadratic equations, the sum of roots = -b/a, product = c/a
Use factoring when possible for simpler solutions

❓ Frequently Asked Questions (FAQ)

What is the discriminant?

The discriminant (Δ = b² - 4ac) tells you the nature of the roots. If Δ > 0, two real roots; Δ = 0, one real root; Δ < 0, two complex roots.

Can a quadratic equation have no solution?

Every quadratic equation has a solution, but it may be complex (imaginary) when the discriminant is negative. Our calculator handles both real and complex roots.

What happens if a = 0 in a quadratic equation?

If a = 0, the equation becomes linear (bx + c = 0). Our calculator automatically handles this case.

How do you check if a number is a root?

Substitute the value into the equation. If the result equals 0, it's a root. For example, if x = 2 is a root of x² - 5x + 6 = 0, then 4 - 10 + 6 = 0.

What is the quadratic formula used for?

The quadratic formula is used to find the roots of any quadratic equation, even when factoring is difficult or impossible. It works for all quadratic equations.

What are complex roots?

Complex roots occur when the discriminant is negative. They are written in the form a ± bi, where i = √-1. For example, x = 2 ± 3i.