A worked example: solving 2x² − 7x + 3 = 0 first
Start with 2x² − 7x + 3 = 0, so a = 2, b = -7, c = 3. The Quadratic Formula Calculator works out the discriminant as (-7)² − 4(2)(3) = 49 − 24 = 25, which is positive, so it returns "Two distinct real roots": root1 = 3 and root2 = 0.5.
The chain: feeding a root into the polynomial calculator
If root1 = 3 is genuinely a solution of 2x² − 7x + 3 = 0, then treating 2x² − 7x + 3 as a polynomial and evaluating it at x = 3 should return exactly zero. Entering coefficients "2, -7, 3" with x = 3 into the Polynomial Calculator returns result = 0. Doing the same with x = 0.5 (the second root) also returns result = 0. As a contrast, evaluating the same polynomial at x = 1, a value that is not a root, returns result = -2 — a clearly nonzero result, which is exactly what should happen at a non-root point.
A second check: x² − 5x + 6 = 0
The same chain works for any quadratic. For x² − 5x + 6 = 0 (a = 1, b = -5, c = 6), the discriminant is (-5)² − 4(1)(6) = 25 − 24 = 1, and the Quadratic Formula Calculator returns root1 = 3 and root2 = 2. Evaluating the polynomial "1, -5, 6" at x = 3 in the Polynomial Calculator returns result = 0, confirming that root once again.
Why this cross-check catches arithmetic mistakes
Solving for a root and evaluating a polynomial at a point are two independent calculations that should agree at exactly one thing — a true root always evaluates the original expression to zero, and a value that only looks close to a root will not. That makes the polynomial calculator a fast, independent way to confirm a root found with the quadratic formula, whether it came from a calculator or by hand.