Example 1
- a = 1
- b = -3
- c = 2
x₁ = 1, x₂ = 2
How many roots when Δ = 9? Two: 1 and 2.
Quadratic equation calculator
Type coefficients a, b and c of ax² + bx + c = 0. The discriminant, Δ = b² - 4ac, says how many real roots you get. With a = 1, b = -3, c = 2 the discriminant is 9 and the roots are x = 1 and x = 2.
When a = 0, the calculator falls back to the linear equation bx + c = 0. A negative discriminant means no real roots.
Results
Enter data and click Calculate.
How it works
A quadratic equation is ax² + bx + c = 0 with a not zero. You look for the x values that make the left side zero. With a = 1, b = -3, c = 2 the equation is x² - 3x + 2 = 0 and the roots are 1 and 2.
The discriminant, written Δ = b² - 4ac, decides the picture. Δ > 0 gives two different x. Δ = 0 gives one double root. Δ < 0 leaves a message that there are no real roots. For 1, -3, 2 you get Δ = 9. For 1, 2, 1 you get Δ = 0 and x = -1.
The formula is x = (-b ± √Δ) / (2a). Plus and minus are the two places on the axis. The calculator rounds to six decimals when the number is not short. The parabola chart shows the same zeros, unless a is zero.
When a = 0 it is no longer a parabola. The calculator then solves bx + c = 0, so x = -c/b, provided b is not zero. With a = 0, b = 2, c = -4 you get x = 2. All-zero a, b, c become either an identity or no solution, in the calculator message.
The fields are coefficient a, coefficient b and coefficient c. A comma and a dot are the same decimal. Type 1, -3 and 2, then click Calculate.
x² + 1 = 0, meaning a = 1, b = 0, c = 1, has discriminant -4 and no real x. x² - 2x + 1 = 0 has x = 1 twice. 2x² - 8 = 0 has x = -2 and x = 2.
Formula
Δ = b² - 4ac, then x = (-b ± √Δ) / (2a)
How to use
- Type a, b and c, for example 1, -3 and 2.
- Click Calculate. The discriminant is 9, x = 1 and x = 2.
- With 1, 2, 1 the discriminant is 0 and you get x = -1, a double root.
- With 1, 0, 1 the discriminant is negative: no real roots.
- When a = 0 the calculator solves bx + c = 0. Plain ax + b = 0 lives on the linear equation card.
Discriminant and two x
ax² + bx + c = 0. Δ = b² - 4ac. For 1, -3, 2 the discriminant is 9 and the roots are x = 1 and x = 2.
- discriminant
- Δ = b² - 4ac. Positive: two x. Zero: one. Negative: no real roots.
- a
- The coefficient of x². Zero falls back to a linear equation. For 1, -3, 2 it keeps the parabola.
- root
- An x that makes the left side zero. With Δ = 9 there are two: 1 and 2.
Examples
Example 2
- a = 1
- b = 2
- c = 1
x = -1
What if Δ = 0? One double root, x = -1.
Example 3
- a = 1
- b = 0
- c = 1
none
Does x² + 1 = 0 have a real x? No. Δ = -4.
Related calculators
Common questions
What is the discriminant for a=1, b=-3, c=2?
Δ = (-3)² - 4·1·2 = 9. Two roots: x = 1 and x = 2.
What does the discriminant mean in a quadratic?
Δ = b² - 4ac. Positive gives two x, zero one double root, negative no real roots.
What x solves x² - 3x + 2 = 0?
1 and 2. You can check: 1 - 3 + 2 = 0 and 4 - 6 + 2 = 0.
What if the discriminant is 0?
One double root x = -b / (2a). For 1, 2, 1 you get x = -1.
Does a negative discriminant give complex x in this calculator?
No. The calculator stays with real numbers and says there are no real roots.
What if a = 0?
It is no longer quadratic. The calculator solves bx + c = 0, so x = -c/b. If b is also 0, you get a no-solution or infinitely-many message.
Does a comma in 1.5 work?
Yes. 1.5 and 1,5 are the same coefficient. The formula does not change with the separator.
Why is there a chart next to the result?
The parabola y = ax²+bx+c shows the same zeros on the axis. When a = 0 it does not draw the chart.
How many x does 2x² - 8 = 0 have?
x = -2 and x = 2. Here b = 0, Δ = 64, roots symmetric about zero.
How is this different from a linear equation?
Here you have x² and a discriminant. On the linear card it stays ax + b = 0 and one x = -b/a, with no parabola.
Knowledge sources
The calculator computes the same formula as the definition below.
Page updated in 2026.