Intermediate Algebra chapter 6

Solving Quadratic Equations by Completing the Square Chapter 6

Square Root Property We have been using only principal square roots, but now we'll start using both the positive and negative square root. The square root property is that if , where a is a real number, then. Example:   Solve  (x - 3)2 = 4.     x - 3 = ±2 Square root property   x = 3 ± 2 Add 3 to both sides   x = 3 + 2 or x = 3 - 2 Two solutions   x = 5, 1 Simplyfying Idea of Completing the Square Given the problem Solve x2 + bx + c = 0, find a number that complets the square of x2 + bx to get to an equivalent equation of the form Solve (x - d)2 = f, that can be solved as in the above example. Example:   Solve  x2 - 2x - 6 = 0.     x2 - 2x = 6 Rewrite equation   x2 - 2x + 1 = 6 + 1 Add square of ½ coefficient of x term to both sides (this is the completing the square step)   (x - 1)2 = 7 Factor left hand side into it's perfect square   x - 1 = ±  Square root property   x = 1 ±  Solve for x   x = 1 +  x = 1 -  Two solutions Steps to Solving a Quadratic Equation by Completing the Square Make the numerical coefficient of the x2 term equal to 1. Rewrite the equation with the constant by itself on the right side of the equation. Take ½ the numerical coefficient of the x term, square it, and add this quantity to both sides of the equation. Factor the trinomial into the square of a binomial. Use the square root property to take the square root of both sides of the equation. Solve for x. Check the solution in the original equation. Note that the variable was called "x" in the above steps, but it can be called by any variable name, not necessarily just x.