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Part III: Conquering Conics and Systems of Equations in Visual Studio .NET
Part III: Conquering Conics and Systems of Equations
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Crossing music and chemistry
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If you took a poll for the most unlikely couple when it comes to professions or interests, music and chemistry would be high on the list It may surprise you, then, that these topics have an amazing similarity Music is made up of octaves of tones eight tones, running from do, re, mi up to do (are you humming the song from The Sound of Music ) and then starting all over again In 1869, Dimitri Ivanovich Mendel ev found that if he arranged the elements in chemistry in order of increasing atomic weight, every eighth element (starting from a given element) had chemical properties similar to the first one He used his discovery to predict that elements existed that scientists hadn t yet discovered What he predicted was the existence of blanks, or missing elements in the pattern Mathematics counting to eight and starting over again guided Pythagoras with the musical scale and Mendel ev with the atomic chart
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Figure 13-9: Counting the intersections of quartic and cubic polynomials
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To solve systems of equations containing two polynomials, you use the substitution method (see 12) Set y equal to y, move all the terms to the left, and simplify: x 4 + 2x 3 - 13x 2 - 14x + 24 = x 3 + 8x 2 - 13x + 4 x 4 + x 3 - 21x 2 - x + 20 = 0 This equation factors into (x + 5)(x + 1)(x 1)(x 4) = 0 (see 8), which gives you the solutions x = 5, 1, 1, and 4 Substituting these values into the cubic (third-degree) equation (you should always substitute into the equation with the smallest exponential values), you get y = 144 when x = 5, y = 24 when x = 1, y = 0 when x = 1, and y = 144 when x = 4 You now have all the points of intersection
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13: Solving Systems of Nonlinear Equations and Inequalities
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Navigating exponential intersections
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Exponential functions are flattened, C-shaped curves when graphed on a grid (I cover exponentials in 10) When exponentials intersect with one another, they usually do so in only one place, creating one common solution Mixing exponential curves with other types of curves produces results similar to those you see when mixing lines and parabolas you may get more than one solution
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Visualizing exponential solutions
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The exponential functions y = 5x and y = 3x have one common solution: They both cross the Y-axis at the point (0, 1) If you let x = 0 in y = 5x, you get y = 50 = 1 Anything to the zero power equals one So, that means substituting 0 for x in y = 3x gives you y = 30 = 1 You get the same number for both equations You know (0, 1) is the only possible solution for the two exponential functions because any other power of 5 and 3 won t give you the same number The numbers 3 and 5 are both prime numbers, and raising them to powers won t create any common solutions You can discover the solution through algebra, through reasoning it out, or by looking at the graphs of the equations The graphs of the exponential functions y = 5x and y = 3x are shown in Figure 13-10a The steeper of the two exponential curves is y = 5x
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Figure 13-10: Graphing systems containing exponential equations
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Figure 13-10b shows the intersections of the line y = 2x + 2 and the exponential function y = 22x + 1 + 2x 1 Due to the complexities of the exponential function, it isn t practical to solve this system of equations algebraically It really takes some help from technology But, if you can determine the solutions by looking at the graphs, take advantage of the situation The two solutions that the line and exponential function appear to have in common where they intersect are at ( 1, 0) and (0, 2), assuming that each notch on each axis moves one unit at a time (you expect only two solutions because the line keeps going on in one direction, and the exponential function just keeps growing, like exponential functions do, and doesn t double back on itself) You can check to be sure that
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