The formula
x = (−b ± √(b² − 4ac)) ÷ 2a
The formula works for any quadratic equation with a ≠ 0, including equations that do not factorise neatly.
Example with steps: x² + 3x − 2 = 0
- Identify
a = 1,b = 3,c = −2 - Calculate the discriminant:
3² − 4(1)(−2) = 17 - Substitute into the formula
- Simplify to
(-3 ± √17)/2 - Show decimal approximations underneath
Why exact answers matter
For x² + 3x − 2 = 0, most calculators answer
x = 0.5615528 and x = −3.5615528. Those
are roundings. The actual roots are:
x = (−3 ± √17) / 2 If your homework asks for exact form, a decimal approximation does not meet that requirement. And if you carry a rounded value into further calculations, the rounding can affect the result.
Simplifying the surd
The square root√72 is not yet in simplified form. Pulling out the largest square factor gives 6√2, and the whole fraction is then
reduced: (8 ± 2√2)/4 becomes
(4 ± √2)/2. Both steps are shown, because simplified exact form is often required in algebra and schoolwork.
For more work with radicals and exponents, see the Square Root & Exponent Calculator.
What the discriminant tells you
The part under the root, b² − 4ac, decides what
kind of answer you are going to get before you finish the calculation.
| Discriminant | Roots | The parabola |
|---|---|---|
| Positive, a perfect square | Two rational roots | Crosses the x-axis twice; the roots can be written as fractions |
| Positive, not a perfect square | Two irrational roots | Crosses the x-axis twice, at surd values |
| Zero | One repeated root | Touches the x-axis at one point |
| Negative | Two complex roots | Never touches the x-axis |
Negative discriminants have answers too
“No real solutions” is true, and it is not the answer. For
x² + 2x + 5 = 0 the roots are
−1 ± 2i — a pair of complex conjugates,
exactly as valid as any other pair. This calculator gives the complex roots instead of stopping at “no real solutions".
When it isn’t a quadratic
If a is zero there is no x² term, so the
quadratic formula cannot be used at all — it would divide by zero.
The equation is linear, with one solution.
Frequently asked questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. It gives both solutions of ax² + bx + c = 0 for any quadratic, whether or not it factorises.
What does the discriminant tell me?
b² − 4ac tells you the kind of roots before you finish: positive means two real roots, zero means one repeated root, negative means two complex roots. If it is a positive perfect square, the roots are rational and the quadratic factorises.
How do I write the answer in exact form?
Keep the square root rather than evaluating it, simplify it by pulling out square factors, then reduce the whole fraction. For example √72 becomes 6√2, and (8 ± 2√2)/4 reduces to (4 ± √2)/2.
What if there are no real solutions?
There are still two solutions, they are just complex. A negative discriminant gives a conjugate pair such as −1 ± 2i, which this calculator shows in full.
Can a be zero?
Not for the quadratic formula, which divides by 2a. If a is zero the equation is linear rather than quadratic, and has a single solution found by rearranging bx + c = 0.