1. WARM UP  Solve using the quadratic formula 1. 2.

2. SOLUTIONS OF QUADRATIC EQUATIONS

3. OBJECTIVES  Determine the nature of the solution of a quadratic equation with real coefficients  Find and use sums and products of solutions of quadratic equations  Find a quadratic equation given its solutions  Solve problems about the nature of solutions to quadratic equations.

4. REAL NUMBER COEFFICIENTS  The quadratic formula can be used when the coefficients are any complex numbers.  Now, we restrict our attention to equations with real number coefficients.  The expression in the quadratic formula is called the discriminant.  From this number we can determine the nature of the solutions of a quadratic equation.

5. THEOREM 8-3 An equation, with a ≠ 0, and all coefficients real numbers has B. two real-number solutions if C. two complex nonreal solutions that are conjugates of each other if A. exactly one real number solution if

6. EXAMPLE 1 Determine the nature of the solutions of a = 9b = -12c = 4 We compute the discriminant. By Theorem 8-3, there is just one solution and it is a real number.

7. EXAMPLE 2 Determine the nature of the solutions of a = 1b = 5c = 8 We compute the discriminant. Since the discriminant is negative, there are two nonreal solutions that are complex conjugates of each other.

8. EXAMPLE 3 Determine the nature of the solutions of a = 1b = 5c = 6 The discriminant is positive, so there are two real solutions

9. TRY THIS… Determine the nature of the solution of each equation 1. 2. 3.

10. SUMS & PRODUCTS OF SOLUTIONS THEOREM 8-4 For the equation the sum of the solutions is and the product of the solutions is. Note that if we express in the form then the sum of the solutions is the additive inverse of the coefficient of the x-term and the product of the solutions in the constant term. We can find the sum and product of the solutions without solving the equation.

11. EXAMPLE 4 Find the sum and product of the solutions of Let and be the solutions to The discriminant is positive, so there are two real solutions

12. TRY THIS… Find the sum and product of the solutions 1. 2.

13. EXAMPLE 5 Find a quadratic equation for which the sum of the solutions is, and the product is We usually write the equation with integer coefficients. Multiplying by 15, the LCD

14. TRY THIS… Find the quadratic equation for which the sum of the solutions is 3 and the product is -1/4.

15. WRITING EQUATIONS FROM SOLUTIONS We can use the principle of zero products to write a quadratic equation whose solutions are known. Find a quadratic equation whose solutions are 3 and -2/5. Multiplying or or Multiplying by the LCM

16. EXAMPLE 7 When radicals are involved, it is sometimes easier to use the properties of the sum and product. Find a quadratic equation whose solutions are and Finding the sum of the solutions Let the solutions be and Finding the product of the solutions

17. TRY THIS… Find the quadratic equation whose solutions are the following:  -4, 5/32. -7, 83. m, n 4. 8, -95.6.