Algebra Calculator
Solve linear equations, quadratic equations and systems of equations with step-by-step working
Linear: ax + b = c — solve for x
x +
=
functions Key Formulas
Linear: x = (c − b) / a
Quadratic: x = (−b ± √(b²−4ac)) / 2a
Discriminant: Δ = b²−4ac; Δ>0 two roots, Δ=0 one root, Δ<0 complex
Cramer's Rule: x = Dₓ/D, y = Dᵧ/D where D = a₁b₂ − a₂b₁
Solution
Enter values and click Solve to see the result here.
What is an Algebra Calculator?
An algebra calculator solves linear and quadratic equations, simplifies expressions, and evaluates polynomial roots. It helps students check their work, understand step-by-step solution methods, and verify answers to algebraic problems quickly.
For linear equations (ax + b = c), the solution is straightforward. For quadratic equations (ax² + bx + c = 0), the quadratic formula is applied to find real or complex roots, the discriminant tells us the nature of solutions, and the vertex of the parabola is computed.
lightbulb Example Calculation
Scenario: Vijay, Class 10 student from Chennai — solving a board exam problem: "A train covers a distance of 300 km. If speed is increased by 10 km/h, it takes 1 hour less. Find original speed." (leads to quadratic: x²+10x−3000=0)
1Discriminant: b²−4ac = 25−24 = 1 (two real roots)
2x = (5 ± √1) / 2 → x = 3 or x = 2
3Verify: (3−2)(3−3) = 0 ✓ and (2−2)(2−3) = 0 ✓
✓ Result: x = 3 and x = 2 (factors: (x−3)(x−2))
help_outlineHow to Use the Algebra Calculator
- Select the equation type using the tabs — Linear (ax + b = c), Quadratic (ax² + bx + c = 0), or System (two equations, two unknowns).
- For Linear: enter the coefficient a, constant b, and right-hand side c; click "Solve Linear Equation" to find x.
- For Quadratic: enter a, b, and c; click "Solve Quadratic" to get both roots via the quadratic formula along with the discriminant value.
- For System of 2 Equations: enter all 6 coefficients (a₁, b₁, c₁ and a₂, b₂, c₂); click "Solve System" — Cramer's Rule is used to find x and y.
- Review the step-by-step working shown below the result to understand the method — useful for verifying homework and exam answers.
Benefits
- Step-by-step working shown for every equation type — learn the method, not just the answer
- Discriminant displayed for quadratics — know nature of roots before solving
- Three equation types in one tool — covers Class 10 to undergraduate algebra
- Handles decimal and negative coefficients accurately
- Cramer's Rule method for simultaneous equations — directly applicable to JEE and board exams
Key Terms
- Linear Equation
- ax + b = c; solution is x = (c − b) / a; exactly one solution (if a ≠ 0)
- Quadratic Equation
- ax² + bx + c = 0; solved by x = (−b ± √Δ) / 2a; up to two roots
- Discriminant (Δ)
- b² − 4ac; Δ > 0 = two real roots, Δ = 0 = one repeated root, Δ < 0 = complex roots
- System of Equations
- Two equations with two unknowns (x, y); solved using Cramer's Rule via determinants
- Cramer's Rule
- x = Dx/D, y = Dy/D where D = a₁b₂ − a₂b₁; fails when D = 0 (no unique solution)
quizFrequently Asked Questions
What does the discriminant tell me about a quadratic equation?
The discriminant (Δ = b² − 4ac) reveals the nature of roots without fully solving: Δ > 0 means two distinct real roots (parabola crosses x-axis twice); Δ = 0 means one repeated real root (parabola touches x-axis at one point); Δ < 0 means complex conjugate roots (parabola never crosses x-axis). Checking the discriminant first is a standard exam technique to avoid unnecessary computation.
When does a system of equations have no solution?
When the two equations represent parallel lines — same coefficient ratios (a₁/a₂ = b₁/b₂) but different right-hand sides — there is no intersection and no solution. The determinant D = a₁b₂ − a₂b₁ equals zero in this case. When D = 0 and the numerator determinants are also zero, the lines coincide and there are infinitely many solutions. Cramer's Rule is undefined when D = 0.
Can this calculator solve equations with fractions or decimals as coefficients?
Yes. Enter decimal coefficients directly (e.g., 1.5 for 3/2, 0.75 for 3/4). The calculator handles all real-number coefficients — integers, decimals, and negatives. For fractions like 2/3, convert to decimal (0.6667) before entering. Results are shown in decimal form and can be converted to fractions manually if needed.
What is the relationship between roots and factors of a quadratic?
If a quadratic ax² + bx + c = 0 has roots r₁ and r₂, it factors as a(x − r₁)(x − r₂). Example: x² − 5x + 6 = 0 has roots x = 2 and x = 3, so it factors as (x − 2)(x − 3). You can verify: sum of roots = r₁ + r₂ = −b/a = 5; product of roots = r₁ × r₂ = c/a = 6. These Vieta's formulas let you cross-check roots quickly without full multiplication.
How is Cramer's Rule applied to solve simultaneous equations?
For the system a₁x + b₁y = c₁ and a₂x + b₂y = c₂: compute the main determinant D = a₁b₂ − a₂b₁; then Dx = c₁b₂ − c₂b₁ (replace the x column with constants); Dy = a₁c₂ − a₂c₁ (replace the y column with constants); finally x = Dx/D and y = Dy/D. Cramer's Rule is especially useful in Class 12 and JEE because it is mechanical and error-resistant once determinants are set up correctly.