QUADRATIC
Quadratic equations, the discriminant decides.
Enter the three coefficients of ax² + bx + c = 0 and get the discriminant Δ = b² − 4ac and both roots: two distinct real roots when Δ > 0, a double root when Δ = 0, conjugate complex roots when Δ < 0. Every step uses the quadratic formula, results are checked with Vieta, and random practice is built in.
Results will appear here
Everything runs locally in your browser
How to use
Enter coefficient a (never 0), b and c, then solve. Δ > 0 gives two distinct real roots, Δ = 0 a repeated root, Δ < 0 a conjugate pair. The working expands the quadratic formula step by step and ends with the Vieta check (sum = −b/a, product = c/a); when the discriminant is a perfect square the roots reduce to simple fractions instead of long decimals.
常见问题
Why can a not be 0?
With a = 0 the quadratic term disappears and the equation becomes linear (bx + c = 0); the formula and discriminant no longer apply. Solve x = −c/b directly - the tool asks you to re-enter.
Where does the discriminant come from?
Completing the square gives (x + b/2a)² = (b² − 4ac) / 4a². The left side is a square; whether the right side can be square-rooted depends entirely on the sign of b² − 4ac: positive gives two values (two roots), zero gives one (repeated), negative has no real square root (go complex). That expression is Δ.
What is Vieta good for?
The sum of the roots is always −b/a and the product c/a, regardless of how messy the roots are. Two uses: verification (every result here shows both), and construction - given a sum S and product P, the two numbers are the roots of x² − Sx + P = 0.
Is any data uploaded?
No. All solving runs locally in the browser.