Completing the Square When the Leading Coefficient Is Not One
The completing-the-square procedure requires the coefficient of the term to be , so that the left side of the equation has the form . When a quadratic equation has a leading coefficient , preliminary steps must be taken to reduce that coefficient to before the standard six-step completing-the-square procedure can begin.
There are two strategies for achieving a leading coefficient of :
Strategy 1 — Factor out the leading coefficient as a GCF. When the leading coefficient divides evenly into all three terms of the trinomial, factor it out as the greatest common factor, then divide both sides of the equation by that coefficient. For example, in , the coefficient divides into , , and , so factoring gives . Dividing both sides by produces , which now has a leading coefficient of .
Strategy 2 — Divide both sides by the leading coefficient. When the leading coefficient does not divide evenly into all terms, divide every term on both sides by the leading coefficient. This produces fraction coefficients for the linear and/or constant terms. The completing-the-square procedure then continues with those fractions, using the same technique already practiced for expressions with fractional linear coefficients.
In both strategies, the goal is the same: transform the equation so that the term stands alone with a coefficient of , enabling the standard completing-the-square steps to proceed.
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