1. WARM UP 1. Simplify 2. Multiply 3. Divide

2. QUADRATIC EQUATIONS INTRODUCTION

3.  Solve equations of the type ax + bx + c =0 OBJECTIVES  Solve a quadratic equation by completing the square.  Solve problems using quadratic equations

4.  The second-degree or quadratic equation models projectile motion by relating the time t an object is in the air to the initial velocity and the height h of an object. INTRODUCTION DEFINITION An equation of the type where a, b, and c are constants and a ≠ 0, is called standard form of a quadratic equation.

5. STANDARD FORM EQUATIONS  Every quadratic polynomial with complex coefficients can be factored into two linear factors. Example Solve This is an equation of the type where c = 0. Quadratic equations in standard form where c = 0 can be easily solved by factoring. Factoring or Using the principle of zero products These numbers check, so the solutions are 0 and. A quadratic equation of this type will always have 0 as one solution. Sometimes it helps to find standard form before factoring.

6. MORE EXAMPLES  Solve (x – 1)(x + 1) = 5(x – 1) Multiplying or Finding standard form These numbers check, so the solutions are 4 and 1. Factoring Using the principle of zero products

7. Solve 1. 2. 3. TRY THIS…

8. MORE EXAMPLES  Consider any quadratic equation in standard form where b = 0, that is, an equation of the form. We can use the multiplication and addition principles to obtain an equation of the form where or Solve Divide both sides by 1/3 or Using the addition property We can abbreviate this as. Sometimes we get solutions that are complex numbers.

9. MORE EXAMPLES Solve Divide both sides by 1/4 or Adding 9 to both sides Finding square roots or

10. Solve 1. 2. 3. TRY THIS…

11. COMPLETING THE SQUARE  The trinomial is the square of a binomial, because. Given the first two terms of a trinomial, we can find the third term that will make it a square. This process is called completing the square. Example: Complete the square for What must be added to to make it a trinomial square? We take half of the coefficient of x and square it. Half of 12 is 6, and 6 is 36. We add 36. is a trinomial square. It is equal to.

12. MORE EXAMPLES  Complete the square for Thus, is a trinomial square. It is Half of ¾ (the coefficient of y) is 3/8.

13. Complete the square 1. 2. 3. 4. TRY THIS…

14. MORE EXAMPLES  We can solve quadratic equations of the form by completing the square. Adding 5 to both sides Solve by completing the square: Adding 1 to complete the square or The numbers check, so they are the solutions. The solutions can be abbreviated as

15. Solve by completing the square 1. 2. TRY THIS…

16. MORE EXAMPLES  For many quadratic equations the leading coefficient is not 1, but we can use the multiplication principle to make it 1. Multiplying both sides by ¼ Solve by completing the square: Take ½ of 3, then add = 9/4. or

17. Solve by completing the square 1. 2. TRY THIS…

18. CH. 8.1 HOMEWORK Textbook pg. 345 #2, 6, 10, 14, 22, 24, & 26