Real and complex roots of the quadratic
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
- Choose Separate coefficients or Plain equation text; each retains its entries.
- Use signed decimals, at most six places. Terms can appear on either side, such as x^2 = 5x - 6.
- Read live roots and rearranged coefficients. Expand discriminant contributions; they classify roots only when a is nonzero.
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 can be explicit or omitted before x in plain mode. Only quadratic, linear and constant terms are supported. Equation text does not expand brackets or accept fractions or scientific notation; numeric coefficient fields retain their existing input behavior.
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.
Rearrange plain equation terms
Entering x^2 = 5x - 6 gives a = 1, b = −5 and c = 6, with roots 2 and 3. Text mode combines like terms on both sides before applying the original solver. An expression without equality means equal to zero.
Check the displayed coefficients to confirm the rearrangement. Each term and combined coefficient must be within ±1,000,000 and six decimal places. Text is bounded to 300 characters and thirty terms; x^2 and x² are accepted, but brackets, other powers and implicit products are not.
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.
Can I type a quadratic instead of entering coefficients?
Yes. Choose Plain equation text and use signed decimal x², x and constant terms, such as x^2 = 5x - 6. The resulting coefficients are displayed.
Sources & further reading
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