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