If the roots of the equations $x^2-bx+c=0$ and $x^2-cx+b=0$ differ by the same quantity, then $b+c$ is equal to $-4$

Let the roots be (p plus q divided by 2) and (p minus q divided by 2), where q is the common difference, giving a sum of roots p equal to b for the first equation and c for the second. Because the roots differ by the same quantity, their product equations b minus c equal to plus or minus q squared are identical in magnitude for both equations. Equating the expressions yields b plus c equals negative four, making the assertion correct. This equality mathematically guarantees the reason as the exact underlying formula for the roots.

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