1.58 Quadratic Square Completion Definitions
Quadratic square completion is a method to rewrite quadratic expressions into a perfect square form, revealing key properties and simplifying solving equations.
Quadratic Square Completion Definitions establishes the vocabulary for two related techniques used to solve quadratic equations that are not factorable over the integers: the direct method of isolating a squared expression and taking its root, and the more general method of deliberately reshaping a quadratic expression into a perfect square trinomial before applying that same root-taking step.
Quadratic Square-Root Method
The quadratic square-root method is a technique for solving a quadratic equation already written, or rewritten, in the form (x − h)² = k, by taking the square root of both sides and accounting for both the positive and negative roots, then isolating the variable. Solving (x − 3)² = 25 using the quadratic square-root method involves taking the square root of both sides to get x − 3 = ±5, then isolating x to obtain the two solutions x = 8 and x = −2.
Quadratic Square Completion
Quadratic square completion is the technique of rewriting a quadratic expression that is not already a perfect square trinomial into an equivalent expression that is one, by adding and subtracting a specifically calculated value, so that the quadratic square-root method can then be applied. Completing the square on x² + 6x + 2 = 0 requires reshaping the expression x² + 6x into a perfect square trinomial, producing the equivalent equation (x + 3)² = 7, which can then be solved directly using the square-root method.
Quadratic Completion Term
The quadratic completion term is the specific value added to (and subtracted from, or moved to the opposite side of) a quadratic expression during square completion, calculated by taking half of the coefficient of the linear term and squaring that result. For the expression x² + 6x, the coefficient of the linear term is 6, half of which is 3, and squaring that gives a quadratic completion term of 9, which is precisely the constant needed to make x² + 6x + 9 factor as the perfect square (x + 3)².
Together, these definitions describe an alternative solving pathway available for any quadratic equation, factorable or not: quadratic square completion uses the quadratic completion term to reshape the expression into a perfect square trinomial, after which the quadratic square-root method extracts the solutions directly by undoing the square.