ganak.appAlgebra Calculator
Free algebra calculator with step-by-step solutions. Solve linear equations, quadratic equations (with graph), systems of two equations, and evaluate polynomials. See discriminant, roots, vertex, factored form, and verification. No sign-up required.
Step-by-Step Solution
See how the answer is calculated
ax + b = cx + d
2x + 3 = 1x + 7
(a − c)x = d − b
(2 − 1)x = 7 − 3
1x = 4
x = (d − b) / (a − c)
x = 4 / 1
x = 4
Substitute x back into both sides
Left: 2(4) + 3 = 11 | Right: 1(4) + 7 = 11
Verified ✓
What Is Algebra?
The foundation of equations, variables, and mathematical reasoning
Algebra is the branch of mathematics that uses letters and symbols to represent numbers and quantities in equations and expressions. It provides tools to describe relationships, find unknown values, and model real-world situations.
At its core, algebra asks: "What value of x makes this statement true?" Whether you are balancing a simple equation or analyzing a system of constraints, the fundamental process is the same — isolate the unknown by applying inverse operations.
Fundamental Equation
If ax + b = 0, then x = −b / a (a ≠ 0)
How to Solve Linear Equations
Step-by-step method for equations of the form ax + b = cx + d
A linear equation has the form ax + b = cx + d where a, b, c, and d are known constants and x is the unknown.
Step-by-step method:
- Move all variable terms to one side: (a − c)x = d − b
- Divide both sides by the coefficient of x: x = (d − b) / (a − c)
- Verify by substituting back into the original equation
Worked Example
3x + 5 = x + 11
(3 − 1)x = 11 − 5 → 2x = 6 → x = 3
Check: 3(3) + 5 = 14, 3 + 11 = 14 ✓
Special cases:
- Identity (a − c = 0, d − b = 0): true for every x — infinitely many solutions
- Contradiction (a − c = 0, d − b ≠ 0): no value of x works — no solution
How to Solve Quadratic Equations
Using the quadratic formula, discriminant, and vertex analysis
A quadratic equation has the standard form ax² + bx + c = 0 (a ≠ 0). It can have two real roots, one repeated root, or two complex conjugate roots.
The Quadratic Formula
x = (−b ± √(b² − 4ac)) / (2a)
The discriminant (Δ = b² − 4ac) determines root type:
Δ > 0
Two distinct real roots
Δ = 0
One repeated real root
Δ < 0
Two complex conjugate roots
Key properties of a parabola:
- Vertex: (−b/2a, f(−b/2a)) — the minimum or maximum point
- Axis of symmetry: x = −b/2a
- Direction: opens upward if a > 0, downward if a < 0
- Y-intercept: (0, c)
Solving Systems of Linear Equations
Finding intersection points with Cramer's Rule
A system of two linear equations in two unknowns has the form:
a₂x + b₂y = c₂
Cramer's Rule:
- Compute the determinant: D = a₁b₂ − a₂b₁
- Solve for x: x = (c₁b₂ − c₂b₁) / D
- Solve for y: y = (a₁c₂ − a₂c₁) / D
Three possible outcomes:
- D ≠ 0: Unique solution — lines intersect at one point
- D = 0, coincident: Infinitely many solutions — same line
- D = 0, parallel: No solution — lines never meet
Evaluating Polynomials
Efficient computation using Horner's method and derivatives
A polynomial is an expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀. Evaluating it at a specific value of x means computing f(x).
Horner's Method (efficient evaluation):
Instead of computing each power separately, nest the multiplications:
This reduces the number of multiplications from O(n²) to O(n) and is more numerically stable.
Derivative (Power Rule):
Each term aₖxᵏ differentiates to k · aₖxᵏ⁻¹. The calculator computes both f(x) and f'(x) automatically.
Common Algebra Mistakes to Avoid
Frequent errors and how to prevent them
Sign errors
Forgetting to flip the sign when moving a term across the equals sign.
Dropping ±
Using only + (or only −) in the quadratic formula instead of ±.
Dividing by zero
Dividing both sides by a variable that might be zero loses solutions.
Incorrect distribution
Forgetting to multiply every term inside parentheses, e.g., 2(x + 3) = 2x + 6, not 2x + 3.
Frequently Asked Questions
Common questions about algebra and solving equations