1 The MEnTe Program Math Enrichment through TechnologyCompleting the Square The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College Click one of the buttons below or press the enter key ©2003 East Los Angeles College. All rights reserved. NEXT EXIT

2 The easiest quadratic equations to solve are of the typewhere r is any constant. Press the right arrow key or the enter key to advance the slides BACK EXIT

3 The solution to is The answer comes from the fact that we solve the equation by taking the square root of both sides Press the right arrow key or the enter key to advance the slides BACK EXIT

4 Since is a constant, we getwhere both sides are positive, but since is a variable it could be negative, yet is positive and is also positive. Press the right arrow key or the enter key to advance the slides BACK EXIT

5 To get around this contradiction we need to insist thatIf we write and is negative, we are saying that a positive number is negative, which it cannot be. To get around this contradiction we need to insist that Press the right arrow key or the enter key to advance the slides BACK EXIT

6 satisfies all possibilities since if is positive we use and if is negative we usePress the right arrow key or the enter key to advance the slides BACK EXIT

7 Thus, we get the solution for We generally change this equation by multiplying both sides by , then we simplify and get Press the right arrow key or the enter key to advance the slides BACK EXIT

8 We generally skip all the intermediate steps for an equation like and getPress the right arrow key or the enter key to advance the slides BACK EXIT

9 For quadratic equations such as we solve and getPress the right arrow key or the enter key to advance the slides BACK EXIT

10 For quadratic equations that are not expressed as an equation between two squares, we can always express them as If this equation can be factored, then it can generally be solved easily. Press the right arrow key or the enter key to advance the slides BACK EXIT

11 If the equation can be put in the form then we can use the square root method described previously to solve it. The solution for this equation is The sign of m needs to be the opposite of the sign used in Press the right arrow key or the enter key to advance the slides BACK EXIT

12 Fortunately the answer is yes!The question becomes: “Can we change the equation from the form to the form ?” Fortunately the answer is yes! Press the right arrow key or the enter key to advance the slides BACK EXIT

13 The procedure for changing is as followsThe procedure for changing is as follows. First, divide by , this gives Then subtract from both sides. This gives Press the right arrow key or the enter key to advance the slides BACK EXIT

14 We pause at this point to review the process of squaring a binomialWe pause at this point to review the process of squaring a binomial. We will use this procedure to help us complete the square. Press the right arrow key or the enter key to advance the slides BACK EXIT

15 Recall that If we let we can solve for to getPress the right arrow key or the enter key to advance the slides BACK EXIT

16 Substituting in we get Using the symmetric property of equations to reverse this equation we getPress the right arrow key or the enter key to advance the slides BACK EXIT

17 Now we will return to where we left our original equationNow we will return to where we left our original equation. If we add to both sides of we get or Press the right arrow key or the enter key to advance the slides BACK EXIT

18 We can now solve this by taking the square root of both sides to getPress the right arrow key or the enter key to advance the slides BACK EXIT

19 is known as the quadratic formulais known as the quadratic formula. It is used to solve any quadratic equation in one variable. We will show how the quadratic equation is used in the example that follows. Press the right arrow key or the enter key to advance the slides BACK EXIT

20 First, we start with an equation Then we change it to From this we getRemember Press the right arrow key or the enter key to advance the slides BACK EXIT

21 We then substitute into the quadratic formula, simplify and get our values for .Press the right arrow key or the enter key to advance the slides BACK EXIT

22 Doing so we get Remember the quadratic equation is and the values are a = 3, b = 11, c = -20 Press the right arrow key or the enter key to advance the slides BACK EXIT

23 This gives us two values for , andPress the right arrow key or the enter key to advance the slides BACK EXIT

24 The equation can be factored into which will give us the same solutions as the quadratic formula. However, the beauty of using the quadratic formula is that it works for ALL quadratic equations, even those not factorable (and even when is negative). Press the right arrow key or the enter key to advance the slides BACK EXIT

25 To review—the steps in using the quadratic formula are as follows: Set the equation equal to zero, being careful not to make an error in signs. Press the right arrow key or the enter key to advance the slides BACK EXIT

26 2. Determine the values of a, b, and c after the equation is set to zero.Remember Press the right arrow key or the enter key to advance the slides BACK EXIT

27 3. Substitute the values of a, b, and c into the quadratic formula.Press the right arrow key or the enter key to advance the slides BACK EXIT

28 4. Simplify the formula after substituting and find the solutions.Press the right arrow key or the enter key to advance the slides BACK EXIT

29 Contact Us At: The MEnTe Program / Title V East Los Angeles College 1301 Avenida Cesar Chavez Monterey Park, CA Phone: (323) Fax: (323) Us At: Our Websites: Press the right arrow key or the enter key to advance the slides BACK EXIT