Quadratic Equation Calculator with Steps
Enter the coefficients of ax2 + bx + c = 0. Get exact and decimal roots, the discriminant, vertex, axis of symmetry, factor form and verification.
Last Updated: July 26, 2026
Solve ax2 + bx + c = 0
Use one field for each coefficient. Fractions, finite decimals, mixed numbers and scientific notation remain exact during the calculation.
Your quadratic solutions will appear here.
Calculation steps
- Enter the three coefficients and solve the equation.
Root details
| Root | Exact form | Decimal form | Check |
|---|---|---|---|
| Solve an equation to see its roots. | |||
Recent calculations
- Your last six results will appear here.
How to Use This Quadratic Equation Calculator
- Write the equation in standard form, ax2 + bx + c = 0.
- Enter the coefficient of x2 in a, the coefficient of x in b and the constant in c. Enter 0 when a term is missing.
- Select how many decimal places you want for approximate roots, vertex coordinates and other nonterminating values.
- Select Solve Quadratic. Review the exact roots, decimals, discriminant, vertex, axis, root relationships and steps.
- Use the root table to confirm the symbolic substitution check. Keep exact fractions or radicals until your final rounding step.
You may enter integers, finite decimals, fractions, mixed numbers or scientific notation. Examples include -7, 0.125, 5/8, -2 1/3 and 4.2e-5. Write 2500 instead of 2,500 because commas are not accepted inside numbers.
What This Quadratic Solver Calculates
This page focuses on one equation already expressed through three coefficients. It normalizes rational coefficients to a smaller equivalent integer equation, calculates the discriminant and applies the quadratic formula. The result separates exact algebraic forms from rounded decimal approximations.
The extra analysis connects the solutions to the related quadratic function y = ax2 + bx + c. You receive the vertex, axis of symmetry, sum and product of roots and a factor form when the roots are rational. These values help you check algebra, interpret a parabola and move between standard, vertex and factored forms.
Use the general Equation Calculator when x appears on both sides or the expression needs expansion first. This specialist page is faster when you already know a, b and c.
Quadratic Standard Form and Coefficients
A quadratic equation in one variable has standard form ax2 + bx + c = 0, where a is not zero. The coefficient a controls the squared term and the direction and width of the related parabola. The coefficient b affects the horizontal position of the vertex. The constant c is the y-intercept of the related function.
Missing terms still need a coefficient. For x2 - 9 = 0, enter a = 1, b = 0 and c = -9. For 4x2 + 7x = 0, enter a = 4, b = 7 and c = 0. If a is zero, the equation is no longer quadratic. The calculator then gives a clearly labeled linear, identity or contradiction result instead of applying the quadratic formula incorrectly.
Quadratic Formula and Discriminant
The quadratic formula solves every quadratic equation with real coefficients. Its discriminant is D = b2 - 4ac. The discriminant sits inside the square root, so its sign determines the type and number of roots.
x = (-b ± √D) / (2a)
- D > 0 produces two distinct real roots.
- D = 0 produces one distinct real root with multiplicity two.
- D < 0 produces two complex conjugate roots.
A positive discriminant does not always mean rational roots. The roots are rational only when the normalized integer discriminant is a perfect square. For example, D = 1 gives rational answers, while D = 8 gives simplified radical answers.
Worked Example: x2 - 5x + 6 = 0
Enter a = 1, b = -5 and c = 6. The discriminant is (-5)2 - 4(1)(6) = 25 - 24 = 1. Substitute these values into the quadratic formula.
x = (5 - 1) / 2 or (5 + 1) / 2
x = 2 or x = 3
The factor form is (x - 2)(x - 3) = 0. The sum of the roots is 5, which equals -b/a. Their product is 6, which equals c/a. Exact substitution gives 22 - 5(2) + 6 = 0 and 32 - 5(3) + 6 = 0.
Vertex and Axis of Symmetry
The related quadratic function has a vertical axis through its vertex. The x-coordinate of the vertex is -b/(2a). Substituting this coordinate into ax2 + bx + c gives the y-coordinate. An equivalent exact formula for the y-coordinate is -D/(4a).
k = -D / (4a)
Vertex = (h, k)
Axis of symmetry: x = h
For x2 - 5x + 6, h = 5/2 and k = -1/4. The vertex is (5/2, -1/4), and the axis is x = 5/2. Because a is positive, the parabola opens upward and this vertex is its minimum point.
Sum, Product and Factor Form
For roots r1 and r2, their sum is -b/a and their product is c/a. These relationships hold for real and complex roots. They provide a quick exact check even when the radical or decimal form looks complicated.
r1 × r2 = c/a
When both roots are rational, the calculator writes an exact factor form based on a(x - r1)(x - r2) = 0. Repeated roots use a(x - r)2 = 0. For irrational roots, the factor card states that no rational linear factor form exists. For complex roots, it states that no real linear factor form exists.
Real, Repeated and Complex Roots
Two real roots are separate x-intercepts of the related parabola. A repeated root occurs where the parabola touches the x-axis at its vertex. Complex roots occur when the parabola has no real x-intercept. They appear as a conjugate pair with the same real part and opposite imaginary parts.
For x2 + 4 = 0, D = -16. The roots are -2i and 2i. The calculator keeps i in the exact answer and gives separate decimal components. It does not describe complex roots as undefined because they are valid solutions over the complex-number system.
Exact Fractions, Radicals and Decimal Values
Finite decimals and written fractions become reduced rational numbers backed by exact integers. The inputs 0.5 and 1/2 therefore produce the same normalized equation and the same exact roots. Mixed numbers are converted before the equation is solved.
A perfect-square discriminant produces rational roots. A non-square positive discriminant produces exact square-root notation and removes detected square factors. A negative discriminant uses i. Decimal values are calculated separately and marked as approximate when rounding changes the exact value. This prevents a rounded number from being copied or displayed as an exact answer.
Linear and Degenerate Input
If a = 0 and b is nonzero, the equation becomes bx + c = 0. The calculator solves this linear fallback as x = -c/b and labels it as non-quadratic. If a = 0, b = 0 and c = 0, every real number satisfies 0 = 0. If only c remains nonzero, no value of x can satisfy c = 0.
These cases are handled before any division by 2a. This prevents a zero leading coefficient from causing a false root or a divide-by-zero failure.
Input Rules and Calculator Limits
Each coefficient field accepts one signed value. A simple fraction may contain decimals or scientific notation on either side of one slash. A mixed number uses a whole number, one space and a proper fraction. Denominators cannot be zero.
Each written numeric part may contain up to 50 significant digits. Scientific exponents must stay from -100 through 100. Normalized coefficients and intermediate exact values have size limits to keep the page responsive. If a value exceeds a limit, the calculator identifies the field and leaves it ready for correction.
Editing a coefficient clears the old result. Changing decimal precision rerenders the decimal fields without changing exact answers or adding another history entry. Recent calculations live only in the current page and disappear on reload. The parser reads a defined numeric grammar and never executes input as JavaScript.
Common Quadratic Formula Mistakes
- Entering a term instead of its coefficient, such as typing 3x for b.
- Forgetting to enter 0 for a missing x or constant term.
- Using a = 0 and still treating the equation as quadratic.
- Calculating b2 without keeping a negative b inside parentheses.
- Changing the sign in -4ac when c is negative.
- Dividing only the square-root term by 2a instead of the full numerator.
- Dropping the plus-or-minus sign and reporting only one root.
- Rounding the discriminant or square root before the final step.
- Calling a negative discriminant an error instead of using complex roots.
- Skipping substitution or the sum-and-product check.
Method and Review Basis
The standard form, zero-product property, completing-the-square method and quadratic formula follow OpenStax College Algebra 2e: Quadratic Equations. The vertex and axis relationships follow OpenStax Quadratic Functions.
The tool normalizes rational coefficients before testing perfect squares and removing detected square factors from radicals. It verifies rational roots by exact substitution. Irrational and complex roots receive a symbolic residual check based on the quadratic formula, so verification does not depend on a rounded decimal.
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Frequently Asked Questions
What does this quadratic equation calculator solve?
It solves ax squared plus bx plus c equals zero from entered coefficients and returns exact and decimal roots with calculation steps.
What should I enter when a term is missing?
Enter 0 for the missing coefficient. For x squared minus 9 equals zero, use a = 1, b = 0 and c = -9.
Can I enter fractions, decimals and mixed numbers?
Yes. Each coefficient accepts integers, finite decimals, simple fractions, mixed numbers and scientific notation.
What is the discriminant?
The discriminant is b squared minus 4ac. Its sign identifies two real roots, one repeated root or two complex roots.
How does the calculator show irrational roots?
It gives an exact square-root form, removes detected square factors and shows a separate rounded decimal approximation.
Does it calculate complex roots?
Yes. A negative discriminant produces two complex conjugate roots written with i and separate decimal forms.
What happens when a equals zero?
The equation is not quadratic. The calculator reports a linear solution, an identity or a contradiction based on b and c.
Does it calculate the vertex?
Yes. It returns the exact vertex coordinates and the axis of symmetry for the related quadratic function.
When does it show a factor form?
It shows an exact factor form when the roots are rational, including the squared factor for a repeated rational root.
How are the roots verified?
Rational roots use exact substitution. Irrational and complex roots use an exact symbolic residual check from the quadratic formula.
Why is my decimal answer marked approximate?
A repeating or irrational value was rounded to the selected number of decimal places, so the exact fraction or radical remains the precise answer.
Are my coefficients sent to a server?
No input is needed by a server to calculate the answer. Parsing, solving and recent history run in the current browser page.
Disclaimer: This calculator provides general mathematical results for learning and checking work. Verify coefficients, notation, domain assumptions and required rounding before using an answer for academic, scientific, financial, engineering or other important decisions.