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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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Four squares by analysis (spoiler) | Comment 4 of 5 |
The rotational symmetry suggests that the square will have its centre at the origin, so that its corners will have coordinates of the form:
(u, v), (-v, u), (-u, -v), (v, -u).

Taking the first two of these points to lie on y = x2 + 8x + 7 gives

v = u2 + 8u + 7              (1)        and       u = v2 - 8v + 7               (2)

Using (1) in (2):             u = (u2 + 8u + 7)2 - 8(u2 + 8u + 7) + 7

which simplifies to:         u4 + 16u3 +70u2 +47u = 0

This quartic has four real roots: 

u =  -7.756981174, -7.7427228955, -0.8157898708, 0

The first three of these give the squares found in previous postings. The fourth gives a square with corners at (0, 7), (-7, 0), (0, -7) and (7, 0).

Iíve been trying to find a way of avoiding the need to solve a cubic to find the three irrational roots, but so far with no success. Any ideas?

  Posted by Harry on 2010-02-27 02:05:46
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