Math

Equation Solver

Solve linear and quadratic equations step by step. Free online equation solver for algebra students.

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What is Equation Solver?

An equation solver finds the values of variables that satisfy mathematical equations. From simple linear equations to complex quadratic and polynomial equations, this tool helps you solve algebraic problems quickly and accurately. Whether you're a student checking homework, a teacher preparing lessons, or a professional solving real-world problems, an equation solver is an indispensable resource. Linear equations (ax + b = 0) have one solution, quadratic equations (ax² + bx + c = 0) can have up to two solutions, and higher-degree polynomials can have multiple solutions. The equation solver uses proven mathematical methods like the quadratic formula, factoring, and numerical approximation to find exact or approximate solutions.

When to Use This Calculator

  • Check algebra homework and exam answers for linear, quadratic, and cubic equations step by step
  • Rearrange physics and chemistry formulas to solve for an unknown variable such as time or velocity
  • Find break-even points and profit thresholds when cost and revenue are modeled by linear or quadratic functions
  • Locate roots of polynomial functions during engineering design and analysis
  • Verify hand-written factoring, completing the square, and other manual solving work
  • Prepare lesson examples and practice problems for students in the classroom

Steps:

  1. Select the type of equation (linear, quadratic, etc.).
  2. Enter the coefficients (a, b, c, etc.).
  3. Click solve to find the solution(s).
  4. View the step-by-step solution process.
  5. Verify the solution by substituting back into the equation.

Formula

Linear: ax + b = 0 → x = -b/a Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a Discriminant: Δ = b² - 4ac Δ > 0: Two real solutions Δ = 0: One real solution Δ < 0: No real solutions (complex)

Use Cases

  • Solving algebra homework and practice problems
  • Finding break-even points in business calculations
  • Determining roots of polynomial functions in engineering
  • Checking manual calculations for accuracy

Key Benefits

  • Solve linear quadratic polynomial step by step
  • Handle multi-variable equation systems
  • Verify solutions catch errors
  • Visualize variable relationships

Pro Tips

  • Check by substituting back
  • Try factoring quadratics first
  • Verify all equations not just one

Common Mistakes to Avoid

  • Not same operation on both sides
  • Losing solutions dividing by zero variable
  • Wrong quadratic formula coefficient signs

Key Terms Explained

Variable: Symbol for unknown value
Quadratic Formula: x equals negative b plus minus root
Discriminant: Determines solution count
System of Equations: Multiple solved together

Related Concepts

  • The quadratic formula solves any quadratic, and our Quadratic Formula Calculator applies it directly with the discriminant shown.
  • Solutions are x-intercepts of the curve, which you can visualize with the Graphing Calculator.
  • Linear systems are solved with Gaussian elimination, the same linear algebra used by the Matrix Calculator.
  • Fractions and decimals in coefficients are handled the same way, and the Fraction Calculator simplifies any that appear in your answer.
  • Linear equations in slope-intercept form y = mx + b relate directly to the Slope Calculator.

Example

Solving 2x² - 5x + 2 = 0: a=2, b=-5, c=2. Discriminant: (-5)² - 4(2)(2) = 25 - 16 = 9. x = [5 ± √9] / 4 = [5 ± 3] / 4. Solutions: x = 2 or x = 0.5. Checking: 2(2)² - 5(2) + 2 = 8 - 10 + 2 = 0 ✓

Interpreting Your Results

Read the solution count together with the discriminant to understand what kind of answer you have. A positive discriminant means two distinct real roots, a zero discriminant means one repeated real root, and a negative discriminant means a conjugate pair of complex roots with no x-axis crossing. For systems, a nonzero determinant signals one unique solution, while a zero determinant means infinitely many or none. Use the step-by-step breakdown to follow each operation, then substitute your answer back into the original equation to confirm it balances.

Frequently Asked Questions

What is the discriminant?
The discriminant (b² - 4ac) determines the nature of quadratic equation solutions. Positive means two real solutions, zero means one repeated solution, and negative means no real solutions (complex numbers).
Can this solver handle equations with no solution?
Yes. For equations like 0x = 5 (no solution) or 0x = 0 (infinite solutions), the solver will indicate the appropriate result.
How do I solve a linear equation with variables on both sides?
Collect the variable terms on one side and the constant terms on the other. For 3x + 6 = 2x + 10, subtract 2x from both sides to get x + 6 = 10, then subtract 6 to get x = 4. Always keep the equation balanced: whatever you do to one side, you must do to the other. Verify by substituting x = 4 back in: 3(4) + 6 = 18 and 2(4) + 10 = 18, so the solution checks.
What does a double (repeated) root mean for a quadratic equation?
A double root happens when the discriminant b² - 4ac equals zero. The quadratic has one solution counted twice, and the graph of the function just touches the x-axis at a single point instead of crossing it. For example, x² + 2x + 1 = 0 factors as (x + 1)² = 0, so the root is x = -1 twice.
Why does a negative discriminant produce complex solutions?
When b² - 4ac is negative, the square root in the quadratic formula has no real value, so the solutions are complex numbers involving the imaginary unit i = √(-1). For x² + 4x + 13 = 0, the discriminant is 16 - 52 = -36, giving x = [-4 ± √(-36)] / 2 = -2 ± 3i. The two complex roots are always a conjugate pair, which reflects the graph of the parabola never crossing the x-axis.
What does the graph of an equation tell me about its solutions?
The real solutions are exactly the x-intercepts where the graph crosses the x-axis. A line crosses once for one root; a parabola that cuts the axis twice has two distinct roots; one that only touches it has a double root; and one that stays entirely above or below the axis has no real roots (only complex ones). The y-intercept is where the graph meets the y-axis, at x = 0.
Can this solver handle cubic and higher-degree polynomials?
Yes. Cubic equations are normalized to standard form and solved with the rational root test together with a numerical approximation method when exact roots cannot be found. Higher-degree polynomials that do not factor cleanly fall back to numerical root-finding, which returns accurate approximate solutions rather than failing.
What is the difference between an identity and a contradiction?
An identity is an equation true for every value of the variable, such as 0x = 0 or x + x = 2x, so it has infinitely many solutions. A contradiction is true for no value, such as 0x = 5, so it has no solution at all. The solver distinguishes these cases and tells you which type you are looking at.
How does the solver handle systems of linear equations?
A 2×2 or 3×3 system is written in the form Ax = b and solved by Gaussian elimination. The determinant of the coefficient matrix tells you the outcome: a nonzero determinant means exactly one unique solution, a zero determinant means either no solution or infinitely many, and the solver reports which case applies.
Can I solve equations with fractions or decimal coefficients?
Yes. You can enter fractions and decimals directly. To simplify by hand, multiply every term by the least common denominator to clear fractions, or multiply decimals by a power of 10 first. For example, 0.5x + 1 = 3 becomes x + 2 = 6 after multiplying by 2, giving x = 4.
When should I factor, complete the square, or use the quadratic formula?
Factor when the quadratic has simple integer factors, as in x² - 5x + 6 = (x - 2)(x - 3), which gives x = 2 or x = 3. Complete the square when you need the vertex form or the solutions involve a perfect square. Use the quadratic formula as a universal method that works for any quadratic, including those with fractions, decimals, or irrational roots.

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