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Quadratic Inequalities I. Homework II. Solving Quadratic Equations We solve quadratic equations by either factoring or using the quadratic formula. We define the discriminant of the quadratic ax2 + bx + c as D = b2 - 4ac We have
How many roots does 1045456564x2 + 3x + 2134534265256 have? III. Quadratic Inequalities Example: x2 - x - 6 > 0 Solution: first we solve the equality by factoring: (x - 3)(x + 2) = 0 Hence x = -2 or x = 3 Next we cut the number line into three regions: x < -2, -2 < x < 3, and x > 3 On the first region (test x = -3), the quadratic is positive, on the second region (test x = 0) the quadratic is negative, and on the third region (test x = 5) the quadratic is positive. Hence our solution is region 1 and region 2. x < -2 or x > 3. We will see how to verify this on a graphing calculator by noticing that y = x2 - x - 6 stays above the x-axis when x < -2 and when x > 3. A demo will be given in class. IV. Applications A 4 ft walkway surrounds a circular flower garden, as shown in the sketch. The area of the walk is 44% of the area of the garden. Find the radius of the garden.
Solution: Area of the walk = pi(4 + r)2 -pi( r)2 = .44(pi)( r)2 Dividing by pi we have, (4 + r)2 - r2 = .44r2 multiplying out, we get, 16 + 8r + r2 -r2 = .44r2 or .44r2 -8r -16 Now use the quadratic formula: a = .44, b = -8, c = -16 so r = (8 +- sqrt(64 - 4(.44)(-16)))/16 or r = 1.1 or -.1 since -.1 does not make sense, we can say that the radius of the garden is 1.1feet.
Example: The profit function for burgers at Heavenly is given by P = 35x - (x2)/25,000,000 - 40,000. Where x represents the number of skiers that come on a given day. How many skiers paying for Heavenly will produce the maximal profit?
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