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Equation Solver Calculator

Solve linear and quadratic equations with step-by-step solutions

Format: ax + b = c (e.g., 2x + 3 = 7, 5x - 2 = 8)

Equation Solver Guide

An equation solver finds exact or decimal solutions for linear, quadratic, and simple systems. It shows key steps—rearranging, simplifying, and applying formulas—so you see how the answer is obtained.

What is Equation Solver?

The equation solver is a study and homework aid for algebra. It handles fractions, radicals, and negative coefficients, and can present results as simplified fractions or decimals.

How to Use the Equation Solver

  1. Choose a type: linear ax + b = 0, quadratic ax^2 + bx + c = 0, or 2x2 system.
  2. Enter coefficients (a, b, c) or matrix entries for systems.
  3. Calculate to get roots/solution and discriminant (quadratic) with steps.
  4. Switch format between fraction and decimal.
  5. Check for undefined cases (e.g., a = 0 in quadratics).

Formulas & Methods

  • Linear: ax + b = 0 -> x = -b/a (a != 0).
  • Quadratic: ax^2 + bx + c = 0 -> x = (-b +- sqrt(b^2 - 4ac)) / (2a) (a != 0).
  • System 2x2:
    a11 x + a12 y = b1
    a21 x + a22 y = b2
    Cramer's rule (determinant D = a11*a22 - a12*a21):
    x = (b1*a22 - a12*b2)/D, y = (a11*b2 - b1*a21)/D (D != 0).

Assumptions & limitations

  • Exact rational arithmetic for fractions; radicals shown when clean.
  • Singular systems (D = 0) have no unique solution; report as none or infinite.

Examples

Example A — Quadratic
x^2 - 5x + 6 = 0: a=1, b=-5, c=6.
Delta = 25 - 24 = 1 -> roots x = (5 +- 1)/2 -> x = 2, 3.

Example B — 2x2 system
2x + 3y = 13, -x + 4y = 5.
D = 2*4 - 3*(-1) = 11 -> x = (13*4 - 3*5)/11 = 2, y = (2*5 - 13*(-1))/11 = 3.

| Type | Key Check | Output | |---|---|---| | Linear | a != 0 | x = -b/a | | Quadratic | Delta sign | Two/one/complex roots | | 2x2 | D != 0 | Unique (x, y) |

Pro Tips & Best Practices

  • Normalize forms first (move all terms to one side) to avoid sign errors.
  • For quadratics, compute Delta before the formula to anticipate root type.
  • Use fraction mode to avoid rounding drift in exact problems.
  • If a = 0 in a "quadratic", switch to the linear solver.

Related Calculators

FAQ

Q: Which equations can this solve?

A: Linear (ax + b = 0), quadratic (ax^2 + bx + c = 0), and 2x2 linear systems. More forms may be supported on the page.

Q: How do I know if a quadratic has real roots?

A: Check the discriminant Delta = b^2 - 4ac. Delta > 0: two reals; Delta = 0: one real (double); Delta < 0: complex conjugates.

Q: Can I see steps?

A: Yes—expand, rearrange, and apply the formula with intermediate values (Delta, sqrt terms).

Q: What about fractions?

A: Inputs like 1/3 are parsed and simplified; results can be shown as fractions or decimals.

Q: Any pitfalls?

A: Watch for a = 0 in quadratic form (it becomes linear), and avoid dividing by zero in system coefficients.

Call to Action

Enter your coefficients and get clean, step-by-step solutions—then switch formats to show exact fractions or decimals in your homework.