Find roots (x-intercepts) of linear, quadratic, cubic, and rational functions
Select Function Type
f(x) = ax + b
f(x)=x +
f(x) = ax^2 + bx + c
f(x)=x^2 +x +
f(x) = ax^3
f(x)=x^3
f(x) = (ax+b)/(cx+d)
f(x)=(x+)/(x+
Result
Root(s)
Derivation
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
AlgebraGraphingPhysicsEngineeringEconomics
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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