Algebra Calculator: Linear, Quadratic, Systems, Factoring & More
Linear Equations
A linear equation ax + b = cx + d has a unique solution x = (d − b)/(a − c) when a ≠ c. If a = c: identity (infinite solutions) or contradiction (no solution). Linear equations model break-even points, rate problems, and any scenario requiring a single unknown.
Quadratic Equations
The quadratic formula x = (−b ± √Δ) / 2a solves ax² + bx + c = 0 for any values. Discriminant Δ = b² − 4ac: positive (two real roots), zero (one double root), negative (complex roots). Vieta's formulas: x₁ + x₂ = −b/a; x₁ × x₂ = c/a. Vertex: (−b/2a, c − b²/4a).
System of Equations (Cramer's Rule)
A 2×2 system a₁x + b₁y = c₁, a₂x + b₂y = c₂ uses determinants: D = a₁b₂ − a₂b₁, Dₓ = c₁b₂ − c₂b₁, Dᵧ = a₁c₂ − a₂c₁. Unique solution: x = Dₓ/D, y = Dᵧ/D when D ≠ 0. D = 0 means parallel (inconsistent) or identical (dependent) lines.
Polynomial Operations & Factoring
Adding/subtracting polynomials: combine like terms (x², x, constants) separately. Multiplying two quadratics via FOIL produces a degree-4 quartic with 5 terms. Factoring reverses multiplication: ax² + bx + c = a(x − r₁)(x − r₂) when Δ ≥ 0, otherwise prime over ℝ.
Polynomial Evaluation (Horner's Method)
Horner's method evaluates f(x) = ax³ + bx² + cx + d as ((ax + b)x + c)x + d using only 3 multiplications — numerically stable and efficient. The derivative f'(x) = 3ax² + 2bx + c shows whether the function is increasing or decreasing at any point.
FAQs
What is the quadratic formula and when do I use it?
The quadratic formula x = (−b ± √(b²−4ac)) / 2a solves any ax²+bx+c = 0. The discriminant Δ = b²−4ac tells the root type: Δ > 0 gives two real roots, Δ = 0 gives one double root, Δ < 0 gives two complex roots.
How does Cramer's rule work for a 2×2 system?
For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, compute D = a₁b₂ − a₂b₁, Dₓ = c₁b₂ − c₂b₁, Dᵧ = a₁c₂ − a₂c₁. Then x = Dₓ/D and y = Dᵧ/D. If D = 0, the system either has no solution (inconsistent) or infinitely many (dependent).
How do I factor a quadratic ax² + bx + c?
Compute Δ = b² − 4ac. If Δ ≥ 0, roots are r₁ = (−b + √Δ)/2a and r₂ = (−b − √Δ)/2a, so the factored form is a(x − r₁)(x − r₂). If Δ < 0, the polynomial is prime over the reals.
What is a linear equation with no solution or infinite solutions?
For ax + b = cx + d: if a = c, it simplifies to b = d. If b = d, it is an identity (infinite solutions). If b ≠ d, it is a contradiction (no solution — parallel lines).
What are Vieta's formulas for a quadratic?
For ax²+bx+c = 0 with roots x₁, x₂: sum = x₁ + x₂ = −b/a; product = x₁ × x₂ = c/a. These let you verify roots without substitution, and construct a quadratic from its roots.
What is Horner's method for polynomial evaluation?
Horner's method evaluates polynomials using nested multiplication. For ax³+bx²+cx+d, compute ((ax+b)x+c)x+d. This requires only 3 multiplications instead of 6 in the naive approach — it's the standard algorithm in calculators.
What is the degree of a polynomial after multiplication?
When multiplying two polynomials, the degree equals the sum of the degrees. Two quadratics (degree 2 × degree 2) produce a degree-4 (quartic) polynomial. Addition/subtraction never increases degree beyond the highest of the operands.
How do I solve a linear equation step by step?
For ax + b = cx + d: (1) Subtract cx: (a−c)x + b = d. (2) Subtract b: (a−c)x = d−b. (3) Divide: x = (d−b)/(a−c). Our calculator shows each step and verifies by substitution.