Graphs

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Simultaneous Quadratics. 17 Equations (i) and (2) give a straight line and the double-branched curve known as the hyper- bola. These intersect at the points P, Q, whose 0\ i^ < cs-r ^^s-r- / Fig. 16. coordinates are (-3, 5) and (5, - 3). These are the only solutions to the system of equations. The auxiliary lines (3) and (4) intersect line (i) in P and Q, and hyperbola (2) in R and 5. \. Xy = 12. 2. X^ -\- y^ = 40. The auxihary equations appearing in the solution are : 3. Jr-f j/ = + 8. 4. X -^y = -^. 5. X—y = ^A, . 6. A'-j = -4. i8 Graphs. The graphs of all these equations are shown in Fig. 17 by the corresponding numbers. The solutions are at the points -P, Q^ R, and 5. Y Fig. 17. The Complex Number. In making general the treatment of quadratics, the complex number becomes necessary. The imaginary unit or z( = V— i) appears in the extraction of the square root of a negative quan- tity. E. G. , V — 4 = V4 ( — I ) = V4 • V — I = ± 2 V— I =±22. a + bi is the type of all complex numbers.

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