Math

Roots Calculator

Find the Roots of Quadratic Equations

Calculate the roots (zeros) of any quadratic equation. The roots are the values of x where the function equals zero, corresponding to the x-intercepts of the parabola. This calculator finds both real and complex roots, displays exact and decimal forms, and shows how the roots relate to the graph.

Quick Tips

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Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Standard Form

Classic quadratic x² − 5x + 6 = 0 with two integer roots (x = 2 and x = 3)

Key values: a = 1 · b = −5 · c = 6

Physics Problem

Projectile height equation −4.9t² + 20t + 1.5 = 0 (when does the ball hit the ground?)

Key values: a = −4.9 · b = 20 · c = 1.5

Complex Roots

Equation x² + 2x + 5 = 0 with no real solutions (discriminant < 0)

Key values: a = 1 · b = 2 · c = 5

Documentation

This calculator is also known as Roots Calculator.

Read the complete guide

Types of Roots

Quadratic equations can have different types of roots:

Rational roots: when the discriminant is a perfect square (roots are fractions or integers)
Irrational roots: when the discriminant is positive but not a perfect square (roots involve square roots)
Repeated root: when the discriminant is zero (the parabola touches the x-axis at one point)
Complex conjugate roots: when the discriminant is negative (roots involve imaginary numbers)

Examples

Identifying Root Types

Compare three equations: x² - 5x + 6 = 0 (rational), x² - 4x + 1 = 0 (irrational), and x² + 2x + 5 = 0 (complex).

With a=1, b=2, c=5, the discriminant is -16 (negative), so the roots are complex: x = -1 ± 2i. These complex conjugate roots mean the parabola never crosses the x-axis.

Key takeaway: The discriminant instantly classifies the root type without needing to solve the full equation.

Understanding Roots Deeply

Strengthen your understanding of polynomial roots:

Every quadratic has exactly two roots (counting multiplicity) by the Fundamental Theorem of Algebra
Use Vieta's formulas to verify: root sum = -b/a and root product = c/a
Rational Root Theorem: if a, b, c are integers and the discriminant is a perfect square, roots are rational
Complex roots always come in conjugate pairs for polynomials with real coefficients

Frequently Asked Questions about Roots Calculator

What are roots of an equation?

Roots (also called zeros or solutions) are the values of x that make the equation equal to zero. Graphically, roots are the x-coordinates where the parabola crosses the x-axis.

What are complex conjugate roots?

Complex conjugate roots come in pairs of the form a + bi and a - bi. They occur when the discriminant is negative, meaning the parabola does not intersect the x-axis.

How can I tell if the roots are rational or irrational?

Check the discriminant: if it is a perfect square (like 0, 1, 4, 9, 16, ...), the roots are rational numbers. If positive but not a perfect square, the roots involve irrational square roots.