60.2 Quadratic Square-Root Method

Quadratic Square-Root Method is the step-by-step procedure for solving a quadratic equation whose variable term already appears, or can be arranged to appear, as a single squared expression, by isolating that expression and applying the square root operation to both sides of the equation.

Squared Expression Isolation

Moving Every Other Term Away

The first step of the method is to isolate the squared expression completely on one side of the equation, moving every other term to the opposite side using addition or subtraction.

Why Full Isolation Is Required

The square root operation can only be applied cleanly to a side of the equation containing nothing but the squared expression; any additional term left on that side would have the square root applied to the entire side incorrectly, rather than to the squared quantity alone.

Exterior Square Coefficient Removal

Dividing Out a Coefficient

If a coefficient multiplies the squared expression, that coefficient is removed by dividing both sides of the equation by it, leaving the squared expression completely alone.

Order Relative to Isolation

This coefficient removal is performed after every other term has already been moved away, since dividing by the coefficient before full isolation would unnecessarily divide terms that are about to be removed anyway.

Opposite-Side Sign Inspection

Checking the Sign of the Isolated Value

Once the squared expression stands alone, the sign of the value on the opposite side of the equation is inspected, since this sign determines which of three distinct solution cases applies.

Why This Inspection Precedes Root-Taking

Determining the sign of n before attempting to take a square root prevents an uninformed attempt to find the square root of a negative number, directing the solver instead toward the correct case for that particular value.

Positive Square Value Case

When the Isolated Value Is Positive

If the value on the opposite side is positive, the equation has two distinct real solutions, found by taking the square root of that positive value.

Why Two Solutions Result

Both a positive and a negative value produce the same positive result when squared, so both must be considered valid solutions to an equation stating that a squared expression equals a positive number.

Zero Square Value Case

When the Isolated Value Is Zero

If the value on the opposite side is exactly zero, the equation has exactly one real solution, since the square root of zero is zero itself.

Why Only One Solution Results

Unlike the positive case, there is no distinct negative counterpart to zero, since zero and its negation are the same value, so the positive and negative branches of the square root collapse into a single solution.

Negative Square Value Case

When the Isolated Value Is Negative

If the value on the opposite side is negative, the equation has no real solution, since no real number produces a negative result when squared.

Why This Case Is Recognized Without Further Work

Because squaring any real number, whether positive, negative, or zero, always produces a result that is zero or positive, an equation requiring a negative squared result is immediately recognized as having no real solution without needing to attempt any further calculation.

Positive and Negative Root Branches

Recording Both Possibilities

For the positive value case, both the positive and negative square roots of the isolated value are recorded together, typically written with a single expression indicating both possibilities.

Necessity of Recording Both Branches

Recording only the positive root would omit a genuine solution to the original equation, since the negative root, once squared, satisfies the equation exactly as validly as the positive root does.

Resulting Linear Equation Resolution

Solving Any Remaining Structure Inside the Root

If the original squared expression was a binomial rather than the variable alone, taking the square root produces a linear equation for each sign branch, which is then solved using standard isolation steps.

Why This Extra Step Is Sometimes Needed

When the squared expression involves more than the bare variable, taking the square root does not immediately isolate the variable itself; the resulting simple equation for each branch must still be solved to reach the final numerical roots.

Square-Root Method Solution Set

Assembling the Final Answer

The solution set for the square-root method consists of the value or values obtained from resolving each sign branch, corresponding to whichever of the positive, zero, or negative cases applied to the isolated value.

Confirming Alignment with the Correct Case

Before finalizing the solution set, it is confirmed to contain two values if the isolated value was positive, one value if it was zero, or no real values at all if it was negative, matching exactly the case identified during sign inspection.