Algebra Calculator

Solve core algebra problems step-by-step: linear equations, quadratic equations, 2×2 systems of equations, and polynomial evaluation. Designed to help you check homework, verify computations in technical work, and understand the algebra behind the answer.

Interactive algebra solver

Choose the type of algebraic problem and enter the coefficients as they appear in your equation.

Linear equation: ax + b = c

Enter the coefficients from your equation written in the form ax + b = c. For example, 3x − 5 = 11 corresponds to a = 3, b = −5, c = 11.

The calculator accepts decimals and scientific notation (e.g. 1.2e-3). For teaching and checking work, always compare the steps with the method required by your instructor or syllabus.

Algebra problems covered by this calculator

Modern algebra courses span a wide range of topics, from simple linear equations to systems of equations and polynomials of higher degree. This tool focuses on core, high-frequency skills that appear in school, college and technical entrance exams:

  • Linear equations in one variable, written in the form \(ax + b = c\)
  • Quadratic equations in the form \(ax^2 + bx + c = 0\)
  • Systems of two linear equations in two unknowns \(x\) and \(y\)
  • Polynomial evaluation \(P(x)\) for polynomials up to degree 4

For each problem type, the calculator shows not only the result but also a clean sequence of algebraic steps. This makes it useful as a learning companion and as a quick verification tool in applied work.

1. Solving linear equations \(ax + b = c\)

A linear equation in one variable has the form:

\[ ax + b = c \]

where \(a\), \(b\) and \(c\) are real numbers and \(a \neq 0\).

The goal is to isolate \(x\). The standard rearrangement steps are:

  1. Subtract \(b\) from both sides: \(ax = c - b\)
  2. Divide by \(a\): \(x = \dfrac{c - b}{a}\)

In the special case where \(a = 0\):

  • If \(b = c\), any real number \(x\) is a solution (infinitely many solutions).
  • If \(b \neq c\), there is no solution (the equation is inconsistent).

2. Quadratic equations and the discriminant

A quadratic equation has the form:

\[ ax^2 + bx + c = 0, \quad a \neq 0 \]

The quadratic formula gives the roots:

\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]

where the discriminant \(\Delta\) is

\[ \Delta = b^2 - 4ac \]

The discriminant controls the nature of the roots:

  • \(\Delta > 0\): two distinct real roots
  • \(\Delta = 0\): one real root with multiplicity 2 (a repeated root)
  • \(\Delta < 0\): two complex conjugate roots

3. Systems of two linear equations

A 2×2 system in variables \(x\) and \(y\) can be written as:

\[ \begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases} \]

The calculator uses the determinant (Cramer’s rule) to classify and solve the system. Define:

\[ D = a_1 b_2 - a_2 b_1 \]
  • If \(D \neq 0\), there is a unique solution: \[ x = \frac{c_1 b_2 - c_2 b_1}{D}, \quad y = \frac{a_1 c_2 - a_2 c_1}{D} \]
  • If \(D = 0\) and the equations reduce to the same line, there are infinitely many solutions.
  • If \(D = 0\) and the lines are parallel but distinct, the system has no solution.

4. Polynomial evaluation with Horner’s method

Given a polynomial of degree at most four:

\[ P(x) = a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 \]

A numerically stable and efficient way to evaluate \(P(x)\) for a given value of \(x\) is Horner’s method:

\[ P(x) = (((a_4 x + a_3)x + a_2)x + a_1)x + a_0 \]

This reduces the number of multiplications and helps maintain numerical accuracy, especially when \(x\) or the coefficients are large in magnitude.

Good practice when using algebra calculators

  • Always rewrite equations into a clean standard form before entering coefficients.
  • Check that coefficients and signs match your original equation.
  • Compare the step-by-step output to the methods used in your course or textbook.
  • Use technology as a way to check and understand, not as a substitute for learning the underlying methods.

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