Equation Solver
Solve linear, quadratic, and cubic equations with every step and formula shown
What Is This Equation Solver?
This tool solves linear, quadratic, and cubic equations in one variable (x), showing every algebraic step and formula used to reach the answer β not just the final result. It automatically detects which type of equation you've entered based on the highest power of x present.
How to Use It
- Pick an example from Quick Insert, or type your own equation.
- Write your equation as a sum of terms using ^ for powers, e.g. "2x^2 + 3x - 2 = 0".
- Click Solve (or press Enter) to see the answer, full working, and formulas used.
- Use Copy Result, Print, or Share Link to save or send your solution.
Equation Types & Formulas Used
Linear (ax + b = 0): x = βb β a
Quadratic (axΒ² + bx + c = 0): x = [βb Β± β(bΒ² β 4ac)] β 2a
Cubic (axΒ³ + bxΒ² + cx + d = 0): Solved via the depressed-cubic substitution x = t β bβ3a, giving tΒ³ + pt + q = 0, then either the trigonometric method (three real roots) or Cardano's formula (one real root + a complex pair), depending on the sign of the discriminant Ξ = β4pΒ³ β 27qΒ².
Explanation of Every Input
Equation
Enter your equation as a sum of monomial terms (coefficient, optional "x", optional "^power") on each side of a single "=" sign. Powers above x^3 or parenthesized/factored expressions aren't supported in this version.
Decimal Places (Advanced Options)
Controls how many decimal places are shown for irrational or non-simple-fraction answers β this doesn't change the underlying calculation, only the display precision.
Understanding Your Results
Real Roots
Values of x where the equation is satisfied and the graph would cross the x-axis.
Complex Roots
Shown as a Β± bi when no real solution exists β common for quadratics with a negative discriminant.
Fraction Answers
Shown alongside the decimal whenever the exact value matches a simple fraction within a reasonable denominator.
No Solution / Infinite Solutions
Special cases when every x term cancels out, leaving either a false or a true statement.
Advantages of Using This Solver
- Shows complete working, not just the final answer.
- Automatically detects linear, quadratic, or cubic β no manual selection.
- Displays both decimal and exact fraction answers where applicable.
- Correctly handles complex roots instead of just reporting an error.
Limitations
- Equations must already be expanded β no parentheses or factored form yet (e.g. (x-1)(x-2)=0 isn't supported).
- Only the variable "x" is recognized.
- Supports up to cubic (x^3) equations only β no quartic or higher in this version.
- Does not graph the equation β use the Graphing Calculator for that.
Common Mistakes to Avoid
- Forgetting the "=" sign or including more than one.
- Using parentheses before they're supported (expand the equation first).
- Using a variable other than x.
- Entering an exponent higher than 3.
Worked Examples
Linear Equation
Equation: 3x + 5 = 20
Answer: x = 5
Moving all terms to one side gives 3x - 15 = 0, then dividing by 3 isolates x directly.
Quadratic with Two Real Roots
Equation: x^2 - 5x + 6 = 0
Answer: x = 2 and x = 3
The discriminant (25 - 24 = 1) is positive, so the quadratic formula gives two distinct real roots.
Quadratic with Complex Roots
Equation: x^2 + 4x + 5 = 0
Answer: x = -2 + i and x = -2 - i
The discriminant (16 - 20 = -4) is negative, so the two roots form a complex conjugate pair.
Cubic with Three Real Roots
Equation: x^3 - 6x^2 + 11x - 6 = 0
Answer: x = 1, x = 2, and x = 3
The cubic discriminant is positive, so the trigonometric method returns three distinct real roots.
Frequently Asked Questions
What types of equations can this solver handle?
This solver handles linear equations (like 3x + 5 = 20), quadratic equations (like x^2 - 5x + 6 = 0), and cubic equations (like x^3 - 6x^2 + 11x - 6 = 0). The equation type is detected automatically from what you type β you don't need to select it manually.
How do I enter my equation?
Type it as a sum of terms with a single "=" sign, using ^ for powers β for example "2x^2 + 3x - 2 = 0" or "3x + 5 = 20". Use the Quick Insert menu for ready-made examples of each type.
Can I use parentheses like (x-1)(x-2)=0?
Not yet β this version requires the equation already expanded into a sum of terms (e.g. "x^2 - 3x + 2 = 0" instead of "(x-1)(x-2) = 0"). Support for factored/parenthesized input may be added in a future update.
What does it mean when I get a decimal AND a fraction?
When the exact answer is a simple fraction (like 1/3 or 5/2), the solver shows both the decimal approximation and the exact fraction side by side. If no simple fraction matches within a reasonable denominator, only the decimal is shown.
Why did my quadratic equation give complex (imaginary) roots?
This happens when the discriminant (bΒ² β 4ac) is negative, meaning the parabola never crosses the x-axis, so there are no real solutions β only a pair of complex conjugate roots, shown in the form a Β± bi.
How does the solver handle cubic equations?
It normalizes the equation, applies a depressed-cubic substitution to remove the squared term, computes the discriminant, and then uses either the trigonometric method (for three real roots) or Cardano's formula (for one real root and a complex conjugate pair) β every step is shown in the solution.
What if my equation has no solution or infinite solutions?
If all the x terms cancel out and you're left with a false statement (like 5 = 0), the solver reports "no solution." If you're left with a true statement (like 0 = 0), it reports "infinitely many solutions" since every value of x satisfies the equation.
Does this solver support variables other than x?
No β this version only recognizes the variable "x". Enter your equation using x as the unknown.
Can I solve equations with higher powers, like x^4?
Not in this version β Phase 1 supports linear, quadratic, and cubic equations only (up to x^3). Higher-degree polynomial support may be added later.
Can I graph the equation I just solved?
Yes β head over to the Graphing Calculator and enter the same expression (set equal to 0, or rearranged as y = ...) to see it plotted visually.
Conclusion
Whether you're checking homework or just need a quick answer, this equation solver shows the full path from problem to solution β every formula, every substitution, every step β for linear, quadratic, and cubic equations alike.