Use this quadratic formula calculator to solve ax^2 + bx + c = 0, see both roots, and understand what the discriminant says about them.
Advanced options
How to use our Quadratic Formula Calculator
- Enter Coefficient a, Coefficient b, and Constant c from your equation written as ax^2 + bx + c = 0.
- Check that Coefficient a is not 0, because a quadratic must have an x^2 term.
- Choose Root format. Pick Complex roots if you want answers in a + bi form when Delta is negative.
- Choose Show steps if you want to see the discriminant, substitution into the formula, and simplification.
- Open Advanced options and set Decimal places if you want a shorter or more detailed decimal answer.
- Click Calculate.
- Read Standard form first to confirm your signs match the equation you meant to enter.
- Use Discriminant (Delta) and Type of roots to see whether the roots are two different real numbers, one repeated real number, or two complex numbers.
- Do a quick sanity check: for real roots, the midpoint of Root x1 and Root x2 should match the Axis of symmetry, and Sum of roots and Product of roots should match your coefficients.
Definitions
Coefficient a: The number multiplying x^2. If it is 0, the equation is not quadratic.
Coefficient b: The number multiplying x.
Constant c: The number with no x attached.
Standard form: The equation written as ax^2 + bx + c = 0.
Discriminant (Delta): The value b^2 - 4ac. It tells what kind of roots the equation has.
Type of roots: A plain-language label such as two distinct real roots, one repeated real root, or two complex roots.
Root x1 and Root x2: The two solutions found from the plus and minus parts of the quadratic formula.
Axis of symmetry: The line x = -b/(2a). For real roots, it is halfway between them.
Sum of roots: The total x1 + x2, which should equal -b/a.
Product of roots: The multiplication x1 times x2, which should equal c/a.
Complex roots: Answers that use i, where i^2 = -1, when the discriminant is negative [2].
Common mistakes and quick fixes
Mistake: Entering an equation that is not in standard form and putting the wrong values into Coefficient a , Coefficient b , or Constant c .
Fix: Rewrite the equation as ax^2 + bx + c = 0 first, then copy each signed number into the matching field and confirm it with Standard form .
Mistake: Typing 0 for Coefficient a .
Fix: Change Coefficient a to the number in front of x^2. If there is no x^2 term, this is not a quadratic and the calculator should not be used for that equation.
Mistake: Losing a negative sign for Coefficient b or Constant c , such as entering 3 instead of -3.
Fix: Include the sign exactly as it appears after you move everything to one side, then recheck Standard form before trusting Root x1 and Root x2 .
Mistake: Choosing Root format as Real roots only and expecting a decimal answer even when Discriminant (Delta) is negative.
Fix: If Delta is below 0, switch Root format to Complex roots to see answers in a + bi form, or keep the real-only setting to show that there are no real roots.
Mistake: Thinking the Axis of symmetry is always a root.
Fix: Treat Axis of symmetry as the x-value halfway between the roots. It matches the roots only when the quadratic has one repeated root.
Mistake: Rounding too early and then thinking the calculator is wrong when checks do not match exactly.
Fix: Increase Decimal places and compare Sum of roots and Product of roots before rounding your final answer.
Limitations & Key Assumptions / Boundary Conditions
- The calculator only solves equations that can be written in standard form as ax^2 + bx + c = 0.
- Coefficient a cannot be 0. If a = 0, the equation is linear or invalid for this method.
- If Root format is set to Real roots only and Delta is negative, the calculator will report no real roots instead of forcing a fake decimal answer.
- Decimal inputs can make Delta extremely close to 0 because of rounding. The root type may be classified using a tiny display tolerance, but the actual root calculation still uses the entered numbers.
- Displayed decimals are rounded to the chosen Decimal places, so small differences can appear when you check results by hand.
- The step breakdown is designed to teach the quadratic formula workflow, not to show every possible algebra shortcut or factoring method.
- For very awkward decimals, repeating decimals, or irrational values, the displayed result is mainly an approximation unless the simplified form is obvious.
Methodology
What the calculator computes
The calculator solves a quadratic equation in standard form, then reports the discriminant, root type, roots, axis of symmetry, sum of roots, and product of roots.
Formulas used
x = (-b +/- sqrt(b^2 - 4ac)) / (2a)
Delta = b^2 - 4ac
x_axis = -b / (2a)
x1 + x2 = -b / a
x1 * x2 = c / a
if Delta < 0, x = (-b / (2a)) +/- (sqrt(-Delta) / (2a)) i
How root type is classified
If Delta is positive, the equation has two distinct real roots. If Delta is 0, it has one repeated real root. If Delta is negative, it has two complex roots [2].
How each output is found
Standard form is rebuilt from your entered values of a, b, and c so you can verify signs.
Discriminant (Delta) is computed first because it tells the calculator which root pattern to use.
Root x1 uses the plus branch of the quadratic formula, and Root x2 uses the minus branch.
Axis of symmetry is found from x = -b/(2a). When roots are real, this is their midpoint.
Sum of roots and Product of roots use Vieta's relationships as quick checks against the coefficients.
If Delta is negative and Root format is Complex roots, the result is written in a + bi form, where i^2 = -1 [2]. If Root format is Real roots only, the calculator shows that there are no real roots instead of inventing decimal values.
Mini example
Suppose a = 1, b = -3, and c = 2.
Delta = (-3)^2 - 4(1)(2) = 9 - 8 = 1
x = (3 +/- sqrt(1)) / 2
x1 = (3 + 1) / 2 = 2
x2 = (3 - 1) / 2 = 1
x_axis = -(-3) / (2 x 1) = 3/2 = 1.5
x1 + x2 = 2 + 1 = 3 = -b/a
x1 * x2 = 2 x 1 = 2 = c/a
This example shows a positive discriminant, so the roots are two different real numbers.
Error handling and display rules
If a = 0, the calculator stops and shows an error because dividing by 2a would cause division by zero, and the equation would not be quadratic [2]. Blank or non-numeric required inputs also produce an error. Very small values that would display as -0 are shown as 0 for clarity.
Assumptions used
The method assumes your equation is already entered in standard form and set equal to 0 [3]. Results may look slightly different from hand work if you round early or choose fewer decimal places.