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Complex Roots of a Quadratic Equation Graphically. 二次方程的复根图解。

Math Teacher Pub Date : 1900-01-01 DOI: 10.5951/mt.63.3.0229

W. G. Brady

{"title":"Complex Roots of a Quadratic Equation Graphically.","authors":"W. G. Brady","doi":"10.5951/mt.63.3.0229","DOIUrl":"https://doi.org/10.5951/mt.63.3.0229","url":null,"abstract":"Fig. 1. Graphic solution of x2 + ax + b = 0 when the roots are complex. C: y = 2 + a + 6. To find the complex roots a ? i? of (1), proceed as follows: Draw the axis of sym metry of the parabola and mark points V the vertex of the parabola and the inter section of the axis of the parabola with the x-axis. Determine Q so that V is the mid point of segment PQ. Draw the line paral lel to the x-axis passing through Q and intersecting the parabola in points R and S. We shall show a and ? to be the x-coor dinates of and | QS respectively. First by the quadratic formula the roots of (1) are (? a ? Va2 ? 46)/2 and, since a2 ? 46 < 0, the roots may be rewritten as ? a/2 ? t'(/46 ? a2)/2. Hence, a = a/2 and ? = (V46 a2)/2. Now the parabola C: y = 2 + a + 6 of figure 1 has standard form y (6 a2/4) = ( + a/2)2. Thus and V will have coordinates (? a/2, 0), (? a/2, 6 ? a2/4) respectively, and Q will have coordinates (? a/2, 26 ? a2/2). Thus a = ? a/2 is verified as the x-coor dinate of (or V or Q). The fact that ? = I QS remains to be verified. The x-coordinate of points R and S will be found by solving the equations of C and RS simultaneously. The line RS will have equation y = 26 ? a2/2. S and R will have, as x-coordinates, the solutions of the equation x2 + ax + 6 = 26 ? a2/2, i.e., a/2 ? (V46 a2)/2 and | QS =V462 a2/2. Then ? = ?(?Si The above method gives the complex roots almost as easily as the conventional graphic method yields the real roots.","PeriodicalId":144125,"journal":{"name":"Math Teacher","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128842421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}