Solve Each Equation by Finding All Roots Calculator

This calculator finds all real and complex roots of quadratic, cubic, and quartic equations. Whether you're solving for x in a simple quadratic equation or finding all roots of a more complex polynomial, this tool provides clear, step-by-step solutions.

How to Use This Calculator

Using this calculator is simple:

Select the type of equation you want to solve (quadratic, cubic, or quartic).
Enter the coefficients for your equation.
Click "Calculate" to find all roots.
Review the results and any additional information provided.

The calculator will display all real and complex roots, along with explanations of how each solution was found.

Types of Equations This Calculator Solves

This calculator can solve three main types of polynomial equations:

Quadratic equations (degree 2)
Cubic equations (degree 3)
Quartic equations (degree 4)

Each type has its own method for finding roots, which the calculator explains in detail.

Quadratic Equations

A quadratic equation has the general form:

ax² + bx + c = 0

The roots can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

The discriminant (b² - 4ac) determines the nature of the roots:

If positive: two distinct real roots
If zero: one real root (a repeated root)
If negative: two complex conjugate roots

Cubic Equations

A cubic equation has the general form:

ax³ + bx² + cx + d = 0

Cubic equations can have one real root and two complex conjugate roots, or three real roots (which may be repeated). The calculator uses Cardano's formula to find exact solutions when possible.

Quartic Equations

A quartic equation has the general form:

ax⁴ + bx³ + cx² + dx + e = 0

Quartic equations can be solved using Ferrari's method, which reduces them to a depressed quartic and then to a resolvent cubic. The calculator handles all cases, including multiple roots and complex solutions.

Complex Roots

When equations have negative discriminants or complex coefficients, the calculator provides roots in the form a + bi, where i is the imaginary unit (√-1). These roots are often pairs of complex conjugates.

Complex roots come in conjugate pairs for real-coefficient polynomials, meaning if a + bi is a root, then a - bi is also a root.

Limitations of This Calculator

While this calculator handles most common polynomial equations, there are some limitations:

Equations with degrees higher than 4 cannot be solved with this calculator.
Very large coefficients may cause precision issues in the results.
The calculator provides exact solutions when possible, but may use numerical approximations for complex cases.

For equations beyond quartic, consider using specialized software or advanced mathematical techniques.

Frequently Asked Questions

What types of equations can this calculator solve?

This calculator can solve quadratic (degree 2), cubic (degree 3), and quartic (degree 4) equations. It finds all real and complex roots for these types of equations.

How do I enter an equation into the calculator?

Select the type of equation you have (quadratic, cubic, or quartic), then enter the coefficients for each term in the equation. The calculator will guide you through the process.

What if my equation has complex roots?

The calculator will display complex roots in the form a + bi, where i is the imaginary unit. It will also explain the nature of the roots and how they were found.

Can this calculator solve equations with more than four terms?

No, this calculator is limited to equations with four or fewer terms (quartic equations). For higher-degree polynomials, you may need more advanced mathematical tools.

How accurate are the solutions provided by this calculator?

The calculator provides exact solutions when possible, using algebraic methods. For complex cases, it may use numerical approximations, which are accurate to within standard floating-point precision.