Root — Definition, Formula & Examples

Root

A solution to an equation of the form f(x) = 0. Roots may be real or complex.

Note: The roots of f(x) = 0 are the same as the zeros of the function f(x). Sometimes in casual usage the words root and zero are used interchangeably.

Example:
The roots of x² – x – 2 = 0 are x = 2 and x = –1.
The equation is satisfied if we substitute either x = 2 or x = –1 into the equation.

Key Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Where:

$x$ = The root(s) of the quadratic equation
$a$ = Coefficient of x², must not be zero
$b$ = Coefficient of x
$c$ = Constant term

Worked Example

Problem: Find the roots of x² − 5x + 6 = 0.

Step 1: Identify the coefficients a, b, and c from the equation.

a = 1, \quad b = -5, \quad c = 6

Step 2: Compute the discriminant, b² − 4ac.

(-5)^2 - 4(1)(6) = 25 - 24 = 1

Step 3: Apply the quadratic formula.

x = \frac{-(-5) \pm \sqrt{1}}{2(1)} = \frac{5 \pm 1}{2}

Step 4: Evaluate both solutions.

x = \frac{5 + 1}{2} = 3 \qquad \text{or} \qquad x = \frac{5 - 1}{2} = 2

Step 5: Verify by factoring: x² − 5x + 6 = (x − 3)(x − 2). Setting each factor to zero confirms the roots.

x - 3 = 0 \Rightarrow x = 3, \qquad x - 2 = 0 \Rightarrow x = 2

Answer: The roots are x = 2 and x = 3.

Another Example

This example shows a case where the discriminant is negative, producing complex (non-real) roots. It demonstrates that not all polynomial equations have real-number solutions.

Problem: Find the roots of x² + 4x + 13 = 0.

Step 1: Identify the coefficients.

a = 1, \quad b = 4, \quad c = 13

Step 2: Compute the discriminant.

4^2 - 4(1)(13) = 16 - 52 = -36

Step 3: Since the discriminant is negative, the roots are complex. Apply the quadratic formula using i = √(−1).

x = \frac{-4 \pm \sqrt{-36}}{2} = \frac{-4 \pm 6i}{2}

Step 4: Simplify each root.

x = -2 + 3i \qquad \text{or} \qquad x = -2 - 3i

Answer: The roots are x = −2 + 3i and x = −2 − 3i.

Frequently Asked Questions

What is the difference between a root and a zero?

A root refers to a solution of the equation f(x) = 0, while a zero refers to a value where the function f(x) equals 0. Mathematically they give the same values, but "root" is associated with equations and "zero" is associated with functions. In practice, the two words are used interchangeably.

How many roots can a polynomial have?

A polynomial of degree n has exactly n roots when counted with multiplicity and including complex roots. This is guaranteed by the Fundamental Theorem of Algebra. For example, a quadratic (degree 2) always has exactly 2 roots, though both roots may be the same number (a repeated root) or they may be complex.

When do you use the quadratic formula to find roots?

Use the quadratic formula whenever you need to solve ax² + bx + c = 0 and factoring is not obvious. It works for every quadratic equation, whether the roots are rational, irrational, or complex. For higher-degree polynomials, other methods like synthetic division or numerical techniques are needed.

Root vs. Zero of a Function

	Root	Zero of a Function
Definition	A value of x that satisfies the equation f(x) = 0	A value of x where the function f(x) outputs 0
Context	Used when discussing equations	Used when discussing functions and their graphs
Graphical meaning	The x-coordinates where the graph crosses or touches the x-axis	Same — the x-intercepts of the graph of f
Can be complex?	Yes — complex roots exist even if they don't appear on a real-number graph	Yes, though complex zeros have no visible point on a standard xy-graph
Typical phrasing	"Find the roots of the equation"	"Find the zeros of the function"

Why It Matters

Finding roots is one of the most fundamental tasks in algebra and appears in nearly every math course from Algebra 1 through Calculus. Roots tell you where a graph crosses the x-axis, which is essential for sketching functions and solving real-world problems involving projectile motion, profit analysis, and engineering design. Understanding roots also leads directly to factoring polynomials, a skill used throughout higher mathematics and science.

Common Mistakes

Mistake: Confusing the roots of an equation with the y-intercept of a graph.

Correction: Roots are the x-values where y = 0 (the graph meets the x-axis). The y-intercept is the y-value where x = 0. For f(x) = x² − 5x + 6, the roots are x = 2 and x = 3, while the y-intercept is f(0) = 6.

Mistake: Assuming every polynomial equation has real roots.

Correction: A polynomial can have complex roots that are not real numbers. For example, x² + 1 = 0 has no real roots; its roots are x = i and x = −i. Always check the discriminant for quadratics to determine the nature of the roots.

Related Terms

Zero of a Function — Equivalent concept viewed from function perspective
Solution — General term for any value satisfying an equation
Equation — The mathematical statement whose roots are sought
Function — The rule f(x) whose zeros correspond to roots
Real Numbers — Number set containing real roots
Complex Numbers — Number set that includes all roots of polynomials
Satisfy — A root satisfies the equation f(x) = 0
Discriminant — Determines the nature and number of real roots