Quadratic Formula Calculator
Solve ax² + bx + c = 0 with the quadratic formula, including real or complex roots, discriminant, vertex, axis of symmetry and parabola direction.
Enter your values
Change any assumption and calculate again to compare results.
What This Quadratic Formula Calculator Calculates
The Quadratic Formula Calculator solves equations in the form ax²+bx+c=0 and explains the type of roots through the discriminant. It also reports the vertex, axis of symmetry and whether the parabola opens upward or downward.
This gives more context than a roots-only answer. The discriminant tells you in advance whether the equation has two real roots, one repeated real root or a complex-conjugate pair. The vertex information connects the equation with its graph.
Formula and Calculation Method
The quadratic formula is x = [−b ± √(b²−4ac)]/(2a). The quantity D=b²−4ac is the discriminant. If D is positive, there are two distinct real roots. If D is zero, both roots are the same. If D is negative, the square root contributes an imaginary component and the roots are a complex-conjugate pair.
The vertex occurs at x=−b/(2a). Substituting that x-value into ax²+bx+c gives the vertex y-coordinate. The sign of a determines whether the graph opens upward or downward.
How to Use This Calculator
- Enter the values requested in the calculator workspace and keep units consistent.
- Select Calculate. The arithmetic runs locally in your browser.
- Review the main answer together with the supporting figures and formula shown beside it.
- Change one assumption at a time when comparing scenarios so you can see which input is affecting the result.
- Use Copy Result, Save Result Image or Share Result when you need a record of the calculation.
The page is designed to show more than a single unexplained number. Supporting values make it easier to verify the arithmetic, compare results with a spreadsheet or textbook, and reproduce the calculation later. When a calculator uses a unit selector, the unit affects the calculation exactly as described in the method section rather than being treated as decoration.
Worked Example
For x²−5x+6=0, a=1, b=−5 and c=6. The discriminant is 1, giving roots x=2 and x=3. The vertex is at x=2.5 with y=−0.25.
Replace the example values with the measurements or numbers that apply to your own problem. The default values are present to make the calculator immediately testable and to show the expected input format.
How to Interpret the Result
Real roots are the x-intercepts of the parabola. A repeated root means the parabola touches the x-axis at its vertex. Complex roots mean the graph never crosses the real x-axis, although the algebraic equation still has two roots in the complex-number system.
If coefficient a is zero, the equation is not quadratic. In that case a linear equation method should be used instead.
Common Search Questions About Quadratic Formula Calculator
How do I calculate this result?
Enter the required values in the calculator above and select Calculate. The result is shown together with supporting values and the formula used, so the arithmetic can be checked instead of treated as a black box.
Can I use this calculator on a phone?
Yes. The v113 workspace is responsive, and on small screens the result opens in the current viewport after calculation.
Does OfficeCalculator.Net upload my inputs?
The arithmetic for this calculator runs locally in the browser. The page does not require an account to perform the calculation.
Important Limitations
The calculator evaluates numeric coefficients and provides decimal roots. It does not simplify radicals symbolically, factor polynomials over exact rational forms, or provide a full step-by-step algebra proof.
Round only after the main calculation when practical. Rounding inputs too early can magnify small differences, especially in formulas that involve powers, roots, logarithms, repeated operations or unit conversions.
Privacy, Mobile Results and Practical Use
The calculator performs its arithmetic in the browser and does not require an account. Values entered into the form are used to generate the result shown on the page. On narrow screens the result panel opens in the current viewport after calculation, so the answer does not get lost below long explanatory content.
Use the calculator as a transparent planning and checking tool. For coursework, engineering work, financial records or any other situation where precision matters, keep the original source values and verify rounding conventions. A calculator can evaluate a formula correctly while still producing an inappropriate real-world answer if the wrong inputs, units or definitions are supplied.
Frequently Asked Questions
What does the discriminant tell me?
D=b²−4ac. Positive D gives two real roots, zero gives one repeated real root, and negative D gives two complex roots.
What if a equals zero?
Then the equation is linear rather than quadratic, so the quadratic formula does not apply.
Why are complex roots paired?
For a quadratic with real coefficients, non-real roots occur as complex conjugates because the coefficients are real.