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Squares joining quadratics (Posted on 2010-02-24) Difficulty: 3 of 5
A and B are points on the graph y=x2+8x+7
C and D are points on the graph y=-x2+8x-7
and ABCD is a square.

Find all possible x-coordinates of point A

See The Solution Submitted by Jer    
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Some Thoughts computer exploration (possible spoiler) | Comment 1 of 5

DEFDBL A-Z
CLS
FOR xdist = 3.1842101291981# TO 3.1842101291982# STEP .00000000000001#
  x1 = -4 + xdist: x2 = 4 - xdist
  y1 = x1 * x1 + 8 * x1 + 7
  y2 = -x2 * x2 + 8 * x2 - 7

  x3 = y1: y3 = -x1
  x4 = y2: y4 = -x2

  y3c = -x3 * x3 + 8 * x3 - 7
  y4c = x4 * x4 + 8 * x4 + 7

  PRINT USING "##.############# "; xdist; y3 - y3c; y4 - y4c
  PRINT USING "##.############# "; x1; y1; x2; y2
  PRINT USING "##.############# "; x3; y3; x4; y4
  PRINT

NEXT xdist

(the bounds of xdist were narrowed down for each run to get as close to zero as possible for the discrepancy between the alternate vertices of the square vs their x,y values computed from the parabolas)

gives

 

 

       x                 y                   x                y  
-0.8157898708018 1.1391941468883  0.8157898708018 -1.1391941468883
 1.1391941468883  0.8157898708018 -1.1391941468883 -0.8157898708018

and A could be either of the points with negative x coordinate, with B being the other.

 

--- later: corrected y-coordinate of first point, and program listing which showed wrong variable being displayed.

Edited on February 24, 2010, 4:11 pm
  Posted by Charlie on 2010-02-24 13:04:26

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