Solving Quadratic Equations by Completing the Square MATH 018 Combined Algebra S. Rook

2 Overview Section 11.1 in the textbook: –Solving quadratic equations by completing the square

Solving Quadratic Equations by Completing the Square

4 Solving Quadratics by Completing the Square Recall the method of factoring used when we encountered quadratic equations in section 6.6 –Obviously works only if the quadratic is not prime Often the case a quadratic cannot be factored We need other methods in this case One such method called Completing the Square involves taking the quadratic and transforming it into a perfect square trinomial –How does a perfect square trinomial (e.g. x 2 + 2x + 1) factor?

Solving Quadratics by Completing the Square (Continued) Recall solving a radical equation like How would we solve ? –Obvious solution is 3, but is there another solution? Thus, when taking an even root, there are TWO solutions indicated by ± 5

Solving Quadratics by Completing the Square (Continued) Now consider the equation x 2 – 2x – 7 = 0 –First move the constant, leaving the x terms on the left side –What constant do we need to add to the x terms so that the resulting trinomial becomes a perfect square trinomial? Take half of the coefficient in front of the x term and then square it This is an equation which means what? –How do we eliminate the square around the x? We are taking the square root of both sides which means? 6

7 Approximate vs. Exact Answers Most radicals are irrational numbers (which means?) Exact answers contain the radicals in their simplest form –Ex: Approximate answers contain a decimal approximations of the radicals –Ex: For this class, unless otherwise indicated, give exact answers

Completing the Square (Example) Ex 1: Solve by completing the square: a)d) b) c) 8

Completing the Square on ax 2 + bx + c Suppose we want to solve 2x 2 – x = 2 –What is different about this example than the others? Before completing the square, the coefficient in front of x 2 MUST be 1! –All terms on both sides must be divided by this coefficient even if fractions result –We cannot complete the square on a hard trinomial Proceed as normal 9

Completing the Square on ax 2 + bx + c (Example) Ex 2: Solve by completing the square: a) b) c) 10

11 Summary After studying these slides, you should know how to do the following: –Solve quadratic equations by completing the square Additional Practice –See the list of suggested problems for 11.1 Next lesson –Solving Quadratic Equations Using the Quadratic Formula (Section 11.2)