61 Quadratic Formula and Discriminant

Quadratic Formula and Discriminant is the study of the single universal formula that solves any quadratic equation directly from its standard-form coefficients, together with the discriminant, a value derived from those same coefficients that predicts the number and nature of the equation's solutions without solving the equation at all.

The Scope of the Quadratic Formula

The quadratic formula solves any quadratic equation ax² + bx + c = 0, with a ≠ 0, regardless of whether the expression factors over the integers, making it the most universally applicable of the quadratic-solving techniques, though often more computationally involved than factoring when factoring is actually available.

Preparing to Apply the Quadratic Formula

Before the formula can be applied, the equation must be arranged in standard form, ax² + bx + c = 0, with every term moved to one side and the expression simplified, so that the values a, b, and c can be correctly identified, including their signs, before substitution into the formula.

Applying the Quadratic Formula

With a, b, and c identified, each value is substituted directly into the formula, and the result is simplified using the order of operations, taking particular care with the sign of -b and with correctly computing the expression under the radical.

This produces two solutions, x = 5/2 and x = -1, computed by evaluating the plus and minus cases of the ± symbol separately, each divided by the same denominator 2a.

The Discriminant

The expression under the radical in the quadratic formula, b² - 4ac, is called the discriminant, and it can be evaluated on its own, before applying the rest of the formula, to determine how many real solutions the equation has and what type they are, without going through the full solving process.

Classifying Solutions Using the Discriminant

The sign of the discriminant classifies a quadratic equation's solutions into three cases. If D is positive, the equation has two distinct real solutions, since the square root of a positive number is a real, nonzero value that produces two different results when added to and subtracted from -b. If D equals zero, the equation has exactly one real solution, a repeated root, since adding and subtracting a square root of zero produces the identical result both times. If D is negative, the equation has no real solutions, since the square root of a negative number is not defined within the real numbers.

Verifying Quadratic Formula Solutions

Solutions obtained through the quadratic formula are verified by substituting each one back into the original equation in standard form and confirming both sides evaluate to the same number, exactly as verifying any equation's solutions; because two solutions are typically produced, each should be checked independently.

Diagnosing Errors in the Quadratic Formula and Discriminant

Common errors in this area include misidentifying a, b, or c due to sign errors, particularly when the equation is not fully arranged in standard form before substitution, forgetting to apply the negative sign correctly to -b at the start of the formula, computing the discriminant incorrectly by mishandling the sign of 4ac, applying the ± symbol to only part of the numerator rather than the entire expression above the fraction bar, and dividing only the radical term by 2a while forgetting to divide the -b term by the same denominator.