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Zero Root Calculator

Find roots (x-intercepts) of linear, quadratic, cubic, and rational functions

Select Function Type

Linear ax+b Quadratic ax^2+bx+c Cubic ax^3 Rational num/den

f(x) = ax + b

f(x)=x +

Root Finding Methods

Linear: ax+b=0 -> x=-b/a

Quadratic: x=[-b +/- sqrt(b^2-4ac)]/(2a)

Cubic ax^3=0 -> x=0 (triple root)

Rational: set numerator=0, exclude denominator=0

Roots (zeros, x-intercepts) are values where f(x)=0. The number of real roots depends on the function type. The fundamental theorem of algebra guarantees n roots for an nth degree polynomial.

⚠ Rational function roots come from the numerator, but exclude values where the denominator is also zero (holes).

What Are Function Roots?

Roots, zeros, and x-intercepts all mean points where f(x)=0. For polynomials, the number of roots equals the degree. The quadratic formula is the most common method. Rational functions require checking numerator zeros while watching for holes.

Linear Root

ax+b=0 -> x=-b/a. One root always. For f(x)=2x-6: root at x=3.

Quadratic Roots

Use discriminant D=b^2-4ac. D>0: 2 real. D=0: 1 repeated. D<0: 2 complex. Use quadratic formula.

Cubic Root

ax^3=0 -> x=0 (multiplicity 3). More complex cubics need factoring or numerical methods.

Rational Root

Set numerator=0, but exclude values making denominator=0. P(x)/Q(x)=0 when P(x)=0 and Q(x)!=0.

Teaching Example: f(x)=x^2-3x+2. Set =0: x=[3 +/- sqrt(9-8)]/2 = [3 +/- 1]/2. Roots: x=2 and x=1. Check: (1)^2-3(1)+2=0, (2)^2-3(2)+2=0.

Applications

Algebra Graphing Physics Engineering Economics

Frequently Asked Questions

How to find roots?▼

Set f(x)=0. Linear: x=-b/a. Quadratic: quadratic formula. Rational: numerator=0.

Quadratic formula?▼

x=[-b +/- sqrt(b^2-4ac)]/(2a). Discriminant D determines number of real roots.

How many roots?▼

Degree determines maximum roots. Linear=1, quadratic=2, cubic=3. Complex roots come in pairs.

Root vs x-intercept?▼

Same meaning. Root/zero = x value where f(x)=0. X-intercept = point (x,0) where graph crosses x-axis.

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