Quadratic Formula Calculator
Quadratic formula calculator
Solve ax² + bx + c = 0 with root classification, step-by-step explanation, and copyable LaTeX code.
InputsQuadratic inputsa, b, c • 8 digitsLive
Solved roots
x1 = 2, x2 = 1
Discriminant: 1. Vertex: (1.5, -0.25). Two distinct real roots.
Live update
Two real roots
x = (-b ± √(b² - 4ac)) / (2a)
Root type
Two real roots
Discriminant
1
Axis of symmetry
1.5
Leading coefficient
1
Formula
x = (-b ± √(b² - 4ac)) / (2a)
Use “Copy LaTeX code” if you need equation markup for notes, LMS, or docs.
Step-by-step derivation
- Step 1: Compute discriminant -> Δ = 1.
- Step 2: Since Δ > 0, roots are real and distinct.
- Step 3: Substitute into quadratic formula -> x1 = 2, x2 = 1.
Formula
x = (-b ± √(b² - 4ac)) / (2a) Δ = b² - 4ac Symbol legend
| Symbol | Meaning | Unit | Copy |
|---|---|---|---|
a | Quadratic coefficient (must be non-zero) | - | |
b | Linear coefficient | - | |
c | Constant term | - | |
D | Discriminant used to classify roots | - | |
x | Equation root values | value of x |
- Compute the discriminant Δ = b² - 4ac first.
- If Δ > 0 there are two distinct real roots; if Δ = 0 there is one repeated real root.
- If Δ < 0, roots are complex conjugates with real part -b/(2a).
Example
Worked example: x² - 3x + 2 = 0
- 1 a = 1, b = -3, c = 2
- 2 Discriminant Δ = (-3)² - 4(1)(2) = 1
- 3 x = (3 ± √1) / 2 gives x = 2 and x = 1
Roots are x = 2 and x = 1.
How
- Enter coefficients a, b, and c from your equation ax² + bx + c = 0.
- Keep a non-zero so the expression remains quadratic.
- Read the live roots and discriminant classification instantly, then copy or reset if needed.
Avoid
- Setting a = 0, which turns the equation into a linear equation.
- Forgetting the ± branch, which drops one root when Δ > 0.
- Treating negative discriminants as invalid instead of complex roots.
FAQ
What does the discriminant tell me?
It determines whether roots are two real, one repeated real, or a complex conjugate pair.
Can coefficients be decimals?
Yes. Decimal and negative coefficients are supported as long as a is not zero.
Does this tool return complex roots?
Yes. For D < 0 it reports roots in a +/- bi form.
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