Homework

2.4 Solving Quadratic EquationsA quadratic equation is of the form ax2 + bx + c = 0

There are several ways to solve a quadratic

1. Factoring

2. Completing the square

3. The quadratic formula

Lets practice one of each kind! setting each factor = to 0

Ex 1) FactoringA company makes holiday themed stuffed animals. For October, their revenue was projected to be R(x) = 21x2 + 75x + 235 where x is the number of each 100 stuffed animals produced. The cost to make the animals is C(x) = 4x2 + 7x + 1200. They sold all the animals, but there were some production problems and the company had a loss of $200 for October. Find the number of stuffed animals they sold.Note: Profit = Revenue Cost

200 = 21x2 + 75x + 235 (4x2 + 7x + 1200)

0 = 17x2 + 68x 735 0 = 17(x2 + 4x 45) 0 = 17(x + 9)(x 5)x = 9, 5(they lost money!)*(remember to subtract WHOLE equation)so 500 stuffed animalsEx 2) Completing the SquareA small rocket is fired into the air from an initial height of 50 ft above ground level. The height, s in feet, of the rocket at any time, t in seconds, is modeled bys = 16t2 + 320t + 50. Find the time the rocket is 25 ft above the ground. (round to nearest second)

s = 25 = 16t2 + 320t + 50 16t2 320t = 25 t2 20t = 1.5625

+ 100+ 100

t = 20.078, 0.078so... 20 secondsEx 3) Quadratic FormulaA rectangular open-top box is to be made by cutting 3 in. squares from each corner of a sheet of metal & folding up the edges. The length of the finished box must be 4 in. longer than twice the width. If the volume of the box must be 150 in3, determine the dimensions of the flat sheet of metal needed to construct the box.

V = l w hh = 3w = xl = 2x + 4V = 150 = (2x + 4)(x)(3) 150 = 6x2 + 12x 0 = 6x2 + 12x 150

l = (2(4.099) + 4) + 3 + 3 = 18.19810.099 in 18.198 in333333334.099w = 4.099 + 3 + 3 = 10.099Not all answers to a quadratic are real. What if in the quadratic formula a negative number appears under the radical?! We have to consider imaginary (use i) answers.Types of Solutions for a Quadraticb2 4ac > 02) b2 4ac = 03) b2 4ac < 0 (positive # under ) 2 real solutions( nothing to + or )

1 real solution(negative # under )

0 real solutions, 2 imag comp conjEx 4) Determine the nature of the solutions of 6x2 2x + 3 = 0(2)2 4(6)(3)= 4 72 = 68 < 02 imaginary complex conjugatesSince complex (imaginary) answers come as a pair, if we know one, we know the other and can write an equation with those solutions! conjugates: a + bi & a bi If a quadratic has solutions a & b, its factors are (x a)(x b) and the quadratic is x2 (a + b)x + (a b) = 0(change sign in middle)sumproductEx 5) Write the equation of a quadratic that has 1 i as one solution.1st solution: 1 i 2nd solution: 1 + i

x2 2x + 2 = 0sum: (1 i) + (1 + i)= 2= 1 + i i i2product: (1 i)(1 + i)= 1 (1) = 2Homework#204 Pg 84 #111 odd, 1527 odd, 4, 14, 31, 33, 35, 37, 38