Quadratic Formula Calculator
Enter the coefficients a, b and c of a quadratic equation to get its roots, the discriminant and the vertex of the parabola, with the working shown step by step.
Last updated November 3, 2026
Result
x₁ = 1, x₂ = 2
Two distinct real roots (discriminant above 0).
| Discriminant (b² − 4ac) | 1 |
|---|---|
| Vertex | (1.5, -0.25) |
| Axis of symmetry | x = 1.5 |
| Opens | Upward |
| y-intercept | 2 |
Show the work
Formula: x = (-b ± sqrt(b² - 4ac)) / 2a
- Equation: x² - 3x + 2 = 0, so a = 1, b = -3, c = 2.
- Discriminant = b² - 4ac = -3² - 4 x 1 x 2 = 1.
- x = (-b ± sqrt(D)) / 2a = (3 ± sqrt(1)) / 2.
- Roots: 1 and 2.
Result: x₁ = 1, x₂ = 2
The quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a
A quadratic equation has the form ax² + bx + c = 0, where a is not zero. The quadratic formula gives every value of x that makes the equation true. Put the three coefficients in, work out the part under the square root, and the plus and minus signs give the two roots.
The part under the square root, b² − 4ac, is called the discriminant. It decides what kind of roots you get before you finish the calculation. A positive discriminant gives two different real roots, zero gives one repeated root, and a negative value gives two complex roots that are conjugates of each other.
The calculator also reports the vertex, the turning point of the parabola y = ax² + bx + c. Its x value is −b ÷ 2a, which is also the axis of symmetry and the midpoint of the two real roots. A positive a opens the parabola upward, and a negative a opens it downward.
See how we calculate for the formulas, sources and review schedule behind every calculator.
What the discriminant tells you
For x² − 3x + 2 = 0 the discriminant is (−3)² − 4 × 1 × 2 = 1, which is positive, so there are two real roots, 1 and 2. For x² − 6x + 9 = 0 it is 36 − 36 = 0, so the single root is 3, where the parabola just touches the x axis.
For x² + 2x + 5 = 0 it is 4 − 20 = −16. The square root of a negative number is not real, so the roots are −1 + 2i and −1 − 2i. The parabola never reaches the x axis, and the real part −1 is still the x value of the vertex.
When a is zero
If a is 0 the equation is no longer quadratic, because the x² term disappears. The formula would divide by zero, so the calculator treats bx + c = 0 as a linear equation and returns x = −c ÷ b. For example 0x² + 2x − 6 = 0 gives x = 3.
If a and b are both 0 the equation reads c = 0. That has no solution when c is not 0, and every number is a solution when c is also 0. The calculator says which case applies instead of showing a misleading number.
Checking the answer
The two roots add up to −b ÷ a and multiply to c ÷ a. For 2x² + 4x − 6 = 0 the roots are 1 and −3. Their sum is −2, which equals −4 ÷ 2, and their product is −3, which equals −6 ÷ 2.
You can also substitute a root back in. With x = −3 the equation gives 2 × 9 − 12 − 6 = 0, so the root is correct. Equations such as 3x² − 5x − 2 = 0 have a root that is not a whole number, −1/3, shown as −0.333333.
Where quadratics show up
A ball thrown upward follows a parabola. If its height is h = −5t² + 20t + 25 meters after t seconds, setting h to 0 gives roots t = −1 and t = 5. The negative root has no physical meaning, so the ball lands after 5 seconds. The vertex at t = 2 gives the peak height of 45 meters.
How to use the Quadratic Formula Calculator
- Rewrite your equation as ax² + bx + c = 0, moving every term to one side.
- Enter a, the x² coefficient, then b and c. Use a minus sign for negative values and 0 for a missing term.
- Read the roots, the discriminant and the vertex under the result.
- Open Show the work to follow the steps, then use Copy, Print or Share.
Worked example
Solve 2x² + 4x − 6 = 0, so a = 2, b = 4 and c = −6. The discriminant is 16 + 48 = 64 and its square root is 8. The roots are (−4 + 8) ÷ 4 = 1 and (−4 − 8) ÷ 4 = −3, and the vertex is at x = −1 with y = −8.
| Step | Calculation | Result |
|---|---|---|
| Discriminant | 4² − 4 × 2 × (−6) = 16 + 48 | 64 |
| Square root | √64 | 8 |
| First root | (−4 + 8) ÷ 4 | 1 |
| Second root | (−4 − 8) ÷ 4 | −3 |
| Vertex | x = −4 ÷ 4, then y = 2 − 4 − 6 | (−1, −8) |
Frequently asked questions
What is the quadratic formula?
It is x = (−b ± √(b² − 4ac)) ÷ 2a. It solves any equation of the form ax² + bx + c = 0 where a is not zero, and the ± sign produces the two roots.
What does a negative discriminant mean?
It means the equation has no real roots. The calculator then shows two complex roots written as a ± bi, and the graph of the parabola does not cross the x axis.
Why does the calculator say a cannot be zero?
It can be zero, but then the equation is linear and the formula would divide by zero. The calculator solves bx + c = 0 instead and tells you that it did so.
Can it solve equations that are not in standard form?
Rewrite the equation first so every term is on one side and the other side is 0. For example, x² = 5x − 6 becomes x² − 5x + 6 = 0, so a = 1, b = −5 and c = 6.
Are my numbers stored or sent anywhere?
No. The calculation runs in your browser and nothing you enter is saved or uploaded unless you copy or share a link yourself.