Understanding your result
Solutions are values of x that make the equation zero. A positive discriminant gives two real roots, zero gives a repeated real root, and a negative value gives a conjugate pair containing i, where i² = −1.
To combine rational coefficients before solving, use the Fraction Calculator and then enter their decimal values within the supported precision.
The formula
The coefficients are real numbers. For a nonzero a, D determines the root type. The vertex coordinates are x = −b/(2a) and y = −D/(4a). When a is zero, the remaining equation is linear rather than quadratic.
Factoring x² − 5x + 6
Set a = 1, b = −5, and c = 6. The discriminant is 25 − 24 = 1. The roots are 2 and 3, matching (x − 2)(x − 3) = 0. The vertex is (2.5, −0.25).
How to use this calculator
- Rearrange the equation so its right side is zero.
- Enter the signed coefficients, including zero for a missing term.
- Calculate and read the root type before using the solutions.
Keep the signs when rearranging
For x² = 5x − 6, subtract 5x and add 6 on both sides. This produces x² − 5x + 6 = 0. Entering b as positive 5 solves a different equation.
A coefficient of one still needs to be entered. Powers other than x², x, and a constant are outside this form; the tool does not parse a typed algebraic expression.
Real roots, complex roots, and numerical precision
A parabola with a negative discriminant does not cross the real x-axis. Its complex roots are valid algebraic solutions, but they are not real intercepts. A repeated root corresponds to the graph touching the axis at its vertex.
Discriminant classification uses exact integer arithmetic for the supported six-decimal coefficients. Roots are numerical approximations displayed with up to twelve significant digits. A rearranged root formula reduces loss of the smaller real root when the two roots differ greatly in size.
Assumptions & limitations
What this calculation assumes
- Coefficients are finite real numbers between −1,000,000 and 1,000,000.
- At most six decimal places are accepted for each coefficient.
What to keep in mind
- The tool does not simplify symbolic radicals or accept complex coefficients.
- Displayed roots are rounded; substituting a rounded root can leave a small residual.
Common questions
What happens when a is zero?
The tool solves bx + c = 0. If b also vanishes, c = 0 is an identity and any nonzero c makes the equation inconsistent.
Why are both roots shown?
A quadratic normally has two roots counted with multiplicity. Both can matter unless the original problem imposes a condition such as a positive length.
What does the imaginary unit i mean?
It is defined by i² = −1. Complex roots appear as a real part plus or minus an imaginary part.