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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.

The quadratic formula gives two answers because it contains the $\pm$ (plus or minus) symbol. This symbol indicates that the expression must be split into two separate calculations: one using addition and one using subtraction.

Conditions: The quadratic formula is being applied to solve an equation.; The expression under the square root (the discriminant) is positive, yielding a real square root.

The quadratic formula gives two answers because it contains the $\pm$ (plus or minus) symbol. This symbol indicates that the expression must be split into two separate calculations: one using addition and one using subtraction.

Conditions: The quadratic formula is being applied to solve an equation.; The expression under the square root (the discriminant) is positive, yielding a real square root.

To solve a quadratic equation using the quadratic formula, first rewrite the equation in standard form $ax^2 + bx + c = 0$ and identify the coefficients $a$, $b$, and $c$. Substitute these values into the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Conditions: The equation must be in standard quadratic form: $ax^2 + bx + c = 0$.; The leading coefficient $a$ must not be zero ($a \neq 0$).

To solve a quadratic equation using the quadratic formula, first rewrite the equation in standard form $ax^2 + bx + c = 0$ and identify the coefficients $a$, $b$, and $c$. Substitute these values into the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Conditions: The equation must be in standard quadratic form: $ax^2 + bx + c = 0$.; The leading coefficient $a$ must not be zero ($a \neq 0$).