SOLUTION: find the quadratic equation when one root is the square of the other

find the quadratic equation when one root 
is the square of the other

There are infinitely many such quadratic 
equations, not just one!

If one root is r and the other r², then 
let's solve it backward:

We end up with 

x = r and x = r²

Before that we'd have

x - r = 0 and x - r² = 0

Before that we'd have

(x - r)(x - r²) = 0

Before that we'd have

x² - r²x - rx + r³ = 0

or 

x² - rx(r-1) + r² = 0

or

x² - r(r-1)x + r³ = 0

Before that we could have had any common
factor k to have divided through by, so we
would have had

kx² - kr(r-1)x + kr² = 0

Here are a bunch of examples of different quadratic 
equations all having one root as the square 
of the other:

 x² - 110x + 1000 = 0 has roots 10 and 100
8x² - 720x + 5832 = 0 has roots 9 and 81
8x² - 48x - 216 = 0 has roots -3 and 9
5x² - 150x - 1080 = 0 has roots -6 and 36
 x² - 30x - 216 = 0 has roots -6 and 36
 x² - 6x - 27 = 0 has roots -3 and 9
2x² - 84x - 686 = 0 has roots -7 and 49
5x² - 150x + 625 = 0 has roots 5 and 25
6x² - 72x + 162 = 0 has roots 3 and 9
9x² - 18x - 72 = 0 has roots -2 and 4
8x² - 336x + 1728 = 0 has roots 6 and 36
7x² - 630x + 5103 = 0 has roots 9 and 81
7x² - 42x + 56 = 0 has roots 2 and 4
4x² - 288x + 2048 = 0 has roots 8 and 64
7x² - 140x - 875 = 0 has roots -5 and 25
8x² - 96x + 216 = 0 has roots 3 and 9
6x² - 12x + 6 = 0 has roots 1 and 1
2x² - 4x + 2 = 0 has roots 1 and 1
2x² - 12x + 16 = 0 has roots 2 and 4
7x² - 504x - 5103 = 0 has roots -9 and 81
9x² - 180x - 1125 = 0 has roots -5 and 25
6x² - 432x - 4374 = 0 has roots -9 and 81
2x² - 12x - 54 = 0 has roots -3 and 9
7x² - 294x - 2401 = 0 has roots -7 and 49
7x² - 294x - 2401 = 0 has roots -7 and 49
4x² - 24x + 32 = 0 has roots 2 and 4
4x² - 120x + 500 = 0 has roots 5 and 25
10x² - 60x - 270 = 0 has roots -3 and 9
5x² - 150x + 625 = 0 has roots 5 and 25
4x² - 288x + 2048 = 0 has roots 8 and 64
2x² - 180x - 2000 = 0 has roots -10 and 100
10x² - 1100x + 10000 = 0 has roots 10 and 100
8x² - 160x - 1000 = 0 has roots -5 and 25
10x² - 200x + 640 = 0 has roots 4 and 16
5x² - 100x - 625 = 0 has roots -5 and 25
3x² - 6x + 3 = 0 has roots 1 and 1
4x² - 80x - 500 = 0 has roots -5 and 25
3x² - 216x + 1536 = 0 has roots 8 and 64
2x² - 4x - 16 = 0 has roots -2 and 4
2x² - 24x - 128 = 0 has roots -4 and 16
 x² - 110x + 1000 = 0 has roots 10 and 100
2x² - 84x - 686 = 0 has roots -7 and 49
 x² - 30x + 125 = 0 has roots 5 and 25
9x² - 270x - 1944 = 0 has roots -6 and 36
3x² - 168x - 1536 = 0 has roots -8 and 64

There are billions of trillions more!

Edwin