Algebra

Algebra Solver

Enter a linear or quadratic equation and get instant step-by-step solutions.

Examples:

Formula Reference

Linear Equation
ax + b = c → x = (c−b)/a
Quadratic Formula
x = (−b ± √(b²−4ac)) / 2a
Discriminant
Δ = b² − 4ac
Vertex Form
y = a(x−h)² + k

How to Solve Algebraic Equations

Algebra is the branch of mathematics that uses symbols (typically letters like x) to represent unknown quantities and establish relationships through equations. Solving an algebraic equation means finding the value(s) of the unknown variable that make the equation true. TheCalcPro's algebra solver handles two fundamental equation types: linear equations and quadratic equations, providing step-by-step solutions so you can learn the method, not just the answer.

The Quadratic Formula

The most important formula in introductory algebra is the quadratic formula, which solves any equation of the form ax² + bx + c = 0:

x = (−b ± √(b² − 4ac)) / 2a

Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term. The expression b² − 4ac is the discriminant (D), which determines whether the equation has two real solutions (D > 0), one repeated solution (D = 0), or two complex solutions (D < 0).

Step-by-Step Example: Solving 2x² + 3x − 5 = 0

  1. Identify coefficients: a = 2, b = 3, c = −5
  2. Compute the discriminant: D = 3² − 4(2)(−5) = 9 + 40 = 49
  3. Since D = 49 > 0, there are two distinct real roots
  4. Apply the formula: x = (−3 ± √49) / (2 × 2) = (−3 ± 7) / 4
  5. Root 1: x = (−3 + 7) / 4 = 4 / 4 = 1
  6. Root 2: x = (−3 − 7) / 4 = −10 / 4 = −2.5

The solutions are x = 1 and x = −2.5. You can verify by substituting each value back into the original equation: 2(1)² + 3(1) − 5 = 2 + 3 − 5 = 0 ✓

Applications of Algebra

Algebraic equations model real-world scenarios in physics (projectile motion), finance (break-even analysis), engineering (circuit equations), and everyday problem solving. For calculus-level differentiation, visit our Calculus Solver. For systems of equations, explore the Equation Solver.

Algebra Solver FAQ

TheCalcPro solves linear equations of the form ax + b = c and quadratic equations of the form ax² + bx + c = 0. Linear equations are solved by isolating the variable through inverse operations. Quadratic equations are solved using the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, with full discriminant analysis.

Use the ^ symbol for powers. For example, enter x^2 for x², 3x^2 + 2x - 5 for a quadratic expression, or 2^10 for 2 raised to the 10th power. The solver automatically parses these notations and identifies the equation type.

The discriminant is the expression D = b² − 4ac found under the square root in the quadratic formula. It determines the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there is exactly one repeated real root; if D < 0, the roots are complex conjugates (involving imaginary numbers). The solver displays the discriminant value and interprets it automatically.

Yes. When fractional or decimal coefficients are detected, the solver processes them with full floating-point precision. For fraction inputs, the engine can multiply through by the least common denominator internally to simplify the working before applying the standard solution method.