60.4 Completing-the-Square Solution

Completing-the-Square Solution is the second half of the completing-the-square process, taking the perfect-square trinomial equation produced during preparation and carrying it forward through rewriting, isolation, and root-taking to reach the final numerical solutions of the original quadratic equation.

Perfect-Square Binomial Rewrite

Converting the Trinomial into a Single Squared Term

The perfect-square trinomial produced during preparation is rewritten as a single binomial expression raised to the second power, using the half-linear-coefficient value already identified.

Why This Rewrite Is Valid

This rewrite is valid because the trinomial was deliberately constructed during preparation to match the exact expanded pattern of a perfect-square binomial, so the two forms represent the same quantity written in different ways.

Completed Square Isolation

Confirming the Squared Term Stands Alone

After the rewrite, the left side of the equation consists of nothing but the single squared binomial, matching the isolated structure required before applying the square-root method.

Connection to the Isolation Requirement

This isolation step confirms that the entire purpose of the preparation and rewrite steps was to reach exactly this required structure, connecting the completing-the-square process directly to the square-root method used next.

Square-Root Method Application

Applying the Square Root to Both Sides

With the squared binomial isolated, the square root operation is applied to both sides of the equation, following the same procedure used whenever a squared expression is isolated on one side.

Consequence of the Squared Binomial

Taking the square root of the squared binomial removes the exponent entirely, leaving the binomial itself on the left side, while the square root of the right side must still be evaluated according to the sign of that value.

Plus-Minus Branch Separation

Splitting into Two Signed Cases

As with the standalone square-root method, both a positive and a negative root of the right-side value must be considered, producing two separate branches to solve.

Necessity of Both Branches

Just as in the standalone square-root method, omitting either branch would discard a genuine solution, since both the positive and negative roots of n satisfy the equation once squared back.

Completion-Method Linear Solutions

Isolating the Variable in Each Branch

Each of the two signed branches is a simple linear equation, solved by moving the constant term to the opposite side to isolate the variable completely.

Combining the Two Isolation Results

Solving both branches produces the two candidate roots of the original quadratic equation, each expressed as the same base value adjusted by adding or subtracting the same square-root quantity.

Completion-Method Solution Set

Assembling the Roots

The solution set is formed by collecting the results of both signed branches together, presented as the complete set of values that satisfy the original quadratic equation.

Equivalence to Other Solving Methods

Although reached through a different sequence of steps, this solution set represents the same roots that would be found for the same equation using factoring, when that equation happens to also be factorable, confirming the consistency of the different solving methods.

Repeated Completion Root Case

When the Right-Side Value Is Zero

If the value on the right side after isolation is exactly zero, both signed branches collapse into the same single value, producing one repeated root rather than two distinct ones.

Recognizing This Case Consistently

This repeated-root outcome matches the same repeated-root situation that arises in factoring when an equation factors into two identical binomials, showing consistency between the two methods even in this special case.

No-Real-Root Completion Case

When the Right-Side Value Is Negative

If the value on the right side after isolation is negative, the square-root step cannot produce a real number, and the original equation is recognized as having no real solution.

Recognizing This Case Consistently

This outcome matches the negative square-value case already established for the standalone square-root method, confirming that reaching this point through the completing-the-square process leads to the identical conclusion about the absence of real solutions.