BIZOMAT MAKEOS · Mathematics
Quadratic and Cubic Equation Solver
What are the roots of this quadratic or cubic equation?
Real and complex roots of a quadratic (ax² + bx + c = 0) or cubic (ax³ + bx² + cx + d = 0) equation, with the discriminant.
Open the calculatorRuns in your browser. No account needed.
How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Equation | choice | Quadratic or cubic. |
| Coefficient a | Leading coefficient (must not be zero). | |
| Coefficient b | Second coefficient. | |
| Coefficient c | Third coefficient. | |
| Coefficient d (cubic) | Constant term of a cubic (ignored for a quadratic). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Equation
- What it is
- Whether the polynomial has x² as its highest power (quadratic, three coefficients) or x³ (cubic, four coefficients). It changes which term each coefficient belongs to.
- Units and format
- Choose one of 2 options from the list: Quadratic: ax² + bx + c = 0; Cubic: ax³ + bx² + cx + d = 0.
- Example
- The form opens with “Cubic: ax³ + bx² + cx + d = 0” selected as a worked example. Choose the option that matches your case.
- Get it right
- For a quadratic, c is the constant term and d is ignored; for a cubic, d is the constant.
What this means: Coefficient a
- What it is
- The number multiplying the highest power of x: x² in a quadratic or x³ in a cubic. Rearrange the equation so one side is zero before reading the coefficients.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must not be zero; if the highest term is missing, choose the lower degree instead.
What this means: Coefficient b
- What it is
- The number multiplying the next power down: x in a quadratic, or x² in a cubic.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -6 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Include the sign, and type 0 if that term is missing.
What this means: Coefficient c
- What it is
- The third coefficient: the constant term of a quadratic, or the number multiplying x in a cubic.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 11 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Its meaning changes with the equation type, so check the chosen form before typing.
What this means: Coefficient d (cubic)
- What it is
- The constant term of a cubic, the number with no x attached once everything is on one side of the equals sign.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -6 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored for a quadratic. Move every term to one side first, so the right-hand side is zero.
Method and formulas
- Quadratic: x = (−b ± √(b² − 4ac)) ÷ 2a.
- Cubic: reduce to t³ + pt + q = 0; three real roots by the trigonometric method when the discriminant is positive, otherwise Cardano’s formula.
- Each root is checked by substitution.
- Example: x³ − 6x² + 11x − 6 = 0 has roots 3, 2, 1.
Limitations
- Floating-point arithmetic; repeated roots may show tiny imaginary parts or differences.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Solve design equations that reduce to a quadratic or cubic
- Find dimensions from area or volume relations
- Check hand algebra
Who it is for
- Engineers
- Technicians
- Engineering students
What you get
- All roots (real and complex)
- Discriminant
- Number of real roots
Interpreting the result
- Pick the physically meaningful root (for example positive and within range).
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 32–34. Topic reference; no source tables or recommended settings are reproduced.
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