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First Principles of Algebra Volume 2; Advanced Course

First Principles of Algebra Volume 2; Advanced Course


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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1912 Excerpt: ...lines, y = a, which are tangent to these curves. 135. Problem. Find the intersection points of the curve y = a? + 3x+5 and the line y = 2. Substituting for y and solving for x we have These results involve the imaginary unit already noticed in 112. Numbers of the type a + 6V--1 are discussed further in 195. Tor the present we will regard such results as merely indicating that the conditions stated by the equations cannot be fulfilled by real numbers. This means that the curve and the line have no point in common, as is evident on constructing the graphs. By proceeding as in 134 we find that the line y = is tangent to the curve y = x2 + 3 x + 5. Clearly all lines y = a, in which a ty, are above this line and hence cut this curve in two points. All such lines for which a Jjt are below the line y = and hence do not meet the curve at all. Solving y = a and jTt: x2 + 3 x + 5 for x by first substituting a for y we have. _-3 V4 a-11 If a the number under the radical sign is positive, and there are two real and distinct values of x. Hence the line and the curve meet in two points. If a ty, the number under the radical sign is negative. Consequently the values of x are imaginary and the line and the curve do not meet. Hence we see that the conclusions obtained from the solution of the equations agree with those obtained from the graphs. 136. From the two preceding problems it appears that it is possible to determine the relative positions of the line and the curve without completely solving the equations. Namely, as soon as y is eliminated and the equation in X is reduced to the form ax + bx + c = 0, we examine b2--4 ac as follows: (1) If b2--4 ac 0, i.e. positive, then the line cuts the curve in two distinct points. (2) If b2--4 ac = 0, ...


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Product Details
  • ISBN-13: 9781130495003
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 34
  • Spine Width: 2 mm
  • Width: 189 mm
  • ISBN-10: 1130495000
  • Publisher Date: 01 Mar 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 82 gr


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