The Quadratic Formula

Standard Form: Before applying the formula, a quadratic equation must be written in the form $ax^2 + bx + c = 0$. In this expression, $a$, $b$, and $c$ are constants (coefficients), and $a$ must not be zero.

The Formula: The solutions for $x$ are given by the expression $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula accounts for both possible values of $x$ through the plus-minus ($\pm$) symbol.

Coefficients: The value $a$ is the coefficient of the $x^2$ term, $b$ is the coefficient of the $x$ term, and $c$ is the constant term. It is vital to include the sign (positive or negative) of each coefficient during substitution.

Derivation: The quadratic formula is derived by applying the method of completing the square to the general equation $ax^2 + bx + c = 0$. This process isolates $x$ by transforming the quadratic into a perfect square trinomial.

The Discriminant: The expression under the square root, $D = b^2 - 4ac$, is known as the discriminant. It determines the nature of the roots: if $D > 0$, there are two distinct real roots; if $D = 0$, there is one repeated real root; and if $D < 0$, there are no real roots.

Symmetry: The formula reveals that the roots are located at an equal distance $\frac{\sqrt{b^2 - 4ac}}{2a}$ from the axis of symmetry, which is defined by the line $x = -\frac{b}{2a}$.

Step 1: Rearrangement: Ensure the equation is in standard form. If the equation is $3x^2 = 5x - 2$, it must be rewritten as $3x^2 - 5x + 2 = 0$ before identifying $a, b,$ and $c$.

Step 2: Identification: List the values clearly: $a = 3, b = -5, c = 2$. Using brackets for negative values is a critical habit to prevent sign errors during calculation.

Step 3: Substitution and Simplification: Plug the values into the formula. Simplify the discriminant first, then take the square root, and finally solve for both the plus and minus cases.

Exact vs. Approximate: If the discriminant is not a perfect square, the answer can be left in surd form (e.g., $\frac{5 \pm \sqrt{13}}{6}$) for exactness, or evaluated as a decimal if the problem specifies a degree of accuracy.

The 'Fraction Bar' Rule: A common mistake is writing the division bar only under the square root. Ensure the line extends under the entire numerator: $-b \pm \sqrt{b^2 - 4ac}$.

Negative Coefficients: When $b$ is negative, $-b$ becomes positive. For example, if $b = -4$, then $-b = -(-4) = 4$. Always use parentheses when squaring negative numbers in a calculator, such as $(-4)^2$.

Accuracy Requirements: Check if the question asks for 'exact form' or 'decimal places'. If it asks for 3 significant figures, do not stop at the surd form; perform the final calculation.

Sanity Check: If the discriminant is negative and you are working within the real number system, double-check your signs for $a$ and $c$. A negative discriminant often indicates a mistake in basic arithmetic if the problem is expected to have real solutions.