61.4 Quadratic Formula Solution

The quadratic formula solution provides a systematic method to find roots of quadratic equations, bridging algebraic theory with practical problem-solving techniques.

Quadratic Formula Solution is the sequence of substitution and simplification steps that carries a prepared quadratic equation, together with its coefficients and discriminant, through the full quadratic formula to reach its final numerical roots.

Negative Linear Coefficient Substitution

Substituting b with Its Sign Reversed

The linear coefficient b is substituted into the formula's numerator with its sign reversed, since the formula places a negative sign directly in front of b.

- b →  if b = - 5,  then - b = 5

Careful Handling When b Is Already Negative

When the extracted value of b is itself negative, reversing its sign produces a positive value; this double negative must be tracked carefully to avoid an incorrect sign entering the numerator.

Discriminant Value Substitution

Placing the Discriminant under the Radical

The previously calculated discriminant value is substituted directly beneath the square root symbol in the formula's numerator.

\sqrt{D} → \sqrt{b^{2} - 4 a c}

Reusing the Already-Calculated Value

Because the discriminant was already computed and classified during preparation, this step reuses that value directly rather than recalculating it, keeping the two stages of the process consistent with one another.

Square-Root Value Simplification

Simplifying the Radical Expression

The square root of the discriminant is simplified as far as possible, either to a whole number if the discriminant is a perfect square, or to a simplified radical expression if it is not.

\sqrt{48} = 4 \sqrt{3}

Why Simplification Happens before Branching

Simplifying the radical before splitting into the two signed branches avoids repeating the same simplification work twice, once for each branch, since both branches use the identical simplified radical value.

Positive Formula Branch

Computing the Addition Branch

The positive branch of the formula is computed by adding the simplified square root value to the negated linear coefficient in the numerator, then dividing the result by twice the leading coefficient.

x = \frac{- b + \sqrt{D}}{2 a}

Independence from the Negative Branch

This branch is computed completely on its own, using addition throughout, without reference to the separate calculation that will be performed for the other branch.

Negative Formula Branch

Computing the Subtraction Branch

The negative branch of the formula is computed by subtracting the simplified square root value from the negated linear coefficient in the numerator, then dividing the result by twice the leading coefficient.

x = \frac{- b - \sqrt{D}}{2 a}

Independence from the Positive Branch

Like the positive branch, this branch is computed entirely on its own, using subtraction throughout, producing a result that is generally different from the positive branch's result.

Formula Fraction Reduction

Simplifying the Resulting Fraction

After each branch produces a numerator and denominator, the resulting fraction is reduced by dividing out any common factor shared between the numerator and the denominator.

Why Reduction Is Performed Last

Reduction is applied only after both the numerator and denominator are fully known, since simplifying too early, before the radical term is resolved, risks reducing incorrectly against only part of the numerator.

Formula Exact Radical Result

Presenting the Final Root in Exact Form

Each branch's final, reduced result is presented as an exact value, either as a whole number, a simplified fraction, or an expression containing a simplified radical, rather than as a decimal approximation.

x = \frac{3 \pm \sqrt{3}}{2}

Consistency with the Emphasis on Exact Values

This exact presentation continues the same preference for exact rather than approximate results established throughout the broader scope of the quadratic formula and discriminant.

Quadratic Formula Solution Set

Assembling Both Branches into the Final Answer

The solution set is formed by collecting the results of the positive and negative branches together, presented as the complete set of roots for the original quadratic equation.

Adjusting for the Discriminant's Classification

The final solution set reflects the classification already determined from the discriminant: two distinct values if the discriminant was positive, a single repeated value if it was zero, or no real values at all if it was negative, with the branch computations above applying only when real roots exist.