1. Dear Power point User, This power point will be best viewed as a slideshow. At the top of the page click on slideshow, then click from the beginning.

2. Chapter 11 Quadratic Equations

3. 11.1 Review of Solving Equation by Factoring 11.2 The Square Root Property and Completing the Square 11.3 The Quadratic Formula Putting It All Together 11.4Equations in Quadratic Form 11.5Formulas and Applications 11 Quadratic Equations

4. The Square Root Property and Completing the Square 11.2 The next method we will discuss for solving quadratic equations is the square root Property. Solve an Equation of the Form Note We can use the square root property to solve an equation containing a squared quantity and a constant.

5. Solve using square root property. Example 1 Solution Square root property. The solution set is {-6, 6}. The check is left to the student. Equivalent way to Solve. Take square root on both sides. The square and root reduce to 1. 1 Square root property gives plus or minus answer.

6. Solve using square root property. Example 2 Solution First step is to isolate the square term, that is on a side by itself. Add 60 on both sides. Square root property. Product rule for radicals. The check is left to the student.

7. Solve using square root property. Example 3 Solution First step is to isolate the square term, that is on a side by itself. Subtract 59 on both sides. Divide by 2 on both sides. Square root property. Recall that. The check is left to the student.

8. Solve an Equation of the Form Solve using square root property. Example 4 Solution The equation has a binomial that is being squared, therefore we can use the square root property. (p-11) squared = constant Square root property This means p-11 = 7 or p-11=-7. Solve equations. or The check is left to the student.

9. Solve using square root property. Example 5 Solution or Square root property. Subtract 5 from each side. Divide by 2. Subtract 11 on both sides. Square root property. Subtract 8 on both sides. The check is left to the student.

10. Did you notice that in Example 5b that a complex number and its conjugate were the solutions to the equations? This will always be true provided that the variables in the equation have real-number coefficients. Note If a + bi is a solution of a quadratic equation having only real coefficients, then a – bi is also a solution. Use the Distance Formula Find the distance between the points (-4,1), (2,5). Example 6 Solution Begin by labeling the points: (-4,1), (2,5). Substitute the values into the distance formula. Substitute values.

11. The next method used to solve a quadratic equation is completing the square. A perfect square trinomial is a trinomial whose factored form is the square of a binomial. Some examples of perfect square trinomials are Complete the Square for an Expression of the Form Finding this perfect square trinomial is called completing the square because the trinomial will factor to the square of a binomial.

12. Example 7 Complete the square for each expression to obtain a perfect square trinomial. Then, Factor. Solution Steps to complete Square 1. Find half of the coefficient of x: 2. Square the results: 3. Add 9 to : 1. Find half of the coefficient of h: 2. Square the results: 3. Add 16 to :

13. Note The coefficient of the squared term must be 1 before you complete the square! Perfect Square TrinomialFactored Form We have seen the following perfect square trinomials and their factored forms. Let’s look at the relationship between the constant in the factored form and the Coefficient of the linear term. Notice that the result of taking half from b is the number that goes in the factored form. This pattern will always hold true and can be helpful in factoring some perfect Square trinomials.

14. Example 8 Complete the square for each expression to obtain a perfect square trinomial. Then, Factor. Steps to complete Square 1. Find half of the coefficient of p: 2. Square the results: 3. Add to : Solution Remember, the result you get in step 1 is number in the factored formed.

15. Solve a Quadratic Equation by Completing the Square

16. Step 2: Get the variable on one side of the equal sign and the constant on the other side: Example 9 Solve by completing the square Solution Step 4: Factor Step 5: Solve using the square root property. Step 1: The coefficient of is already 1. Step 3: Complete the square: Subtract 9 on both sides. or

17. Example 10 Solve by completing the square Solution Step 4: Factor Step 5: Solve using the square root property. –SEE NEXT SLIDE. Step 1: Since the coefficient of is not 1, divide the whole equation by 9. Step 2: The constant is on a side by itself. Rewrite the left side of the equation. Step 3: Complete the square: Get a common Denominator.

18. Step 5: Solve using the square root property. Square Root Property Quotient Rule for Radicals Simplify the radical. Subtract 5/6. OR