Alan

I just did it by inspection.

It was fairly obvious that 3^2 = 9 and 2^3 = 8

From this it is clear that x = 3^2 and y = 2^2

The slope of the quadratic at any point is given by 6x - 15. At the minimum the slope is zero, so this must occur when x = 15/6 → 5/2

At this point y = 1, so:   1 = 3*(5/2)^2 - 15*5/2 + k

You can rearrange this to find k.

Suppose the perpendicular line has the equation y = m*x + c

The slope m will be -1/(-2), and because the line goes through (1, -4) you have:

-4 = m*1 + c. ⇒ -4 = 1/2 + c. which allows you to find c.

We can see by plotting a graph that geno is correct:

Here's a graph to illustrate the result:

The symmetry of the parabola dictates that if the vertex is at (5, 3) and it goes through (2, 0) then it must also go through (8, 0).

So there are three conditions from which to find the three unknowns:

3 = 25a + 5b + c.   (1)

0 = 4a + 2b + c.     (2)

0 = 64a + 8b + c.   (3)

From (2) we get.  c = -4a - 2b.  (4)

Put this into (1) and (2).  

3 = 21a + 3b.     (5)

0 = 60a + 6b.     (6)

Divide (5) by 3 and (6) by 6:

1 = 7a + b.    (7)

0 = 10a + b.  (8)

From (8) we get. b = -10a.   (9)

Put this into (7)

1 = -3a.  or:               a = -1/3

Put this into (9):        b = 10/3

Put these into (4):     c = -16/3

See if this image helps

x = 9

y = 4

As follows:

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Note that the prime factorisation of 10^10 is 2^10*5^10

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