If the two roots of ax²+bx+c=0 are α and β, you can know in advance that α+β = −b/a and αβ = c/a without calculating the roots directly with the quadratic formula. Change the coefficients a, b, and c with the sliders, then check whether the sum and product of the actual roots always match these formulas.
Why does this relationship hold? Suppose ax²+bx+c = a(x−α)(x−β). Expanding the right-hand side and dividing both sides by a to match the coefficient of x² gives x²+(b/a)x+(c/a) = x²−(α+β)x+αβ. For the two polynomials to be equal, their coefficients of x and their constant terms must match, so −(α+β) = b/a and αβ = c/a. Rearranging gives α+β = −b/a and αβ = c/a.
Once you know this relationship, you can solve problems such as "What is the sum of the two roots?" and "What is the product of the two roots?" just by looking at the coefficients, without finding the roots directly.