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The expression under the square root, $b^2 - 4ac$, is evaluated to determine the value added to or subtracted from $-b$ in the numerator. Its square root directly influences the final numerical values of the two solutions.

Conditions: The quadratic formula is being applied.; The coefficients $a$, $b$, and $c$ are known.

The expression under the square root, $b^2 - 4ac$, is evaluated to determine the value added to or subtracted from $-b$ in the numerator. Its square root directly influences the final numerical values of the two solutions.

Conditions: The quadratic formula is being applied.; The coefficients $a$, $b$, and $c$ are known.