2. ACTIVITY 12 Quadratic Equations (Section 1.3, pp. 97-105)

3. Quadratics and Intergal Domains A quadratic equation is an equation of the form: ax 2 + bx + c = 0 where a, b, and c are real numbers with a ≠ 0. Zero-Product Property: For any A,B ∈ R: AB = 0 if and only if A = 0 or B = 0.

4. Example 1:  Solve the following equations by factoring: x 2 − 7x + 12 = 0 So by the zero-product property we have or (but in our case and) That is that x = 3 and x = 4 and

5. Completing the Square: To make a perfect square out of x 2 + bx, add the square of half the coefficient of x, that is (b/2) 2. Thus:

6. Example 2: Solve each equation by completing the square: We need (4/2) 2 =4 but we have a -6

8. Example 3: Find all solutions of each equation:

9. and

10. A closer look at roots For the polynomial 3x 2 +7x+4 we saw that x=-1 and x=-4/3 are roots. Does this help us factor this polynomial? Let look as a simpler example: suppose that we have x 2 -1 Since this is the difference of two squares we know that it factors to So if we wish to find the roots we need to set our polynomial equal to zero And solve

11. Consequently If c is a root then (x-c) is a factor and conversely. For the polynomial 3x 2 +7x+4 we saw that x=-1 and x=-4/3 are roots. So (x-(-1)) = x+1 and (x-(-4/3)) = x+(4/3) are both factors THIS IS VERY CLOSE TO 3x 2 +7x+4 WE JUST NEED TO MULTIPLY EVERYTHING BY 3! Consequently,

13. and

14. LCD = x

15. The Discriminant: The discriminant D of the quadratic equation ax 2 + bx + c = 0, where a ≠ 0, is: D = b 2 − 4ac 1. If D > 0 the eq. has 2 distinct real roots. 2. If D = 0 the eq. has exactly 1 real root. 3. If D < 0 the eq. has no real roots.

16. Example 4: Use the discriminant to determine how many real roots each equation has. Do not solve the equation. A= 3 B= -5 C= 1 Thus, there are TWO real solutions!

17. x 2 = 6x − 10 A= 1 B= -6 C= 10 Thus, there are NO real solutions!

18. Example 5: Find all values of k that ensure that the equation kx 2 + 36x + k = 0 has exactly one root (solution). A= k B= 36 C= k

19. Example 7 (Falling-Body Problem):  An object is thrown straight upward at an initial speed of 400 ft/s. From Physics, it is known that, after t seconds, it reaches a height of h feet given by the formula: h = − 16t 2 + 400t. When does the object fall back to ground level?

20. (b) When does it reach a height of 1,600 ft?

21.  (d) How high is the highest point the object reaches? Feet