Question

the sum of a number and its square is one and a half times their difference. what is the number?

Answer 1

direct translation of the English to Math:

x + x^2 = 1.5|x^2 - x|


case1:
times 2
2x + 2x^2 = 3x^2 - 3x
x^2 - 5x = 0
x(x-5) = 0
x = 0 or x = 5

or

case2:
x+x^2 = 1.5(x-x^2)
times 2
2x + 2x^2 = 3x - 3x^2
5x^2 - x = 0
x(5x-1) = 0
x = 0 or x = 1/5

so x = 0 , (the trivial case)
or x = 5 or x = 1/5

check:
the x = 0 case we can check mentally
if x = 5
sum of squares = 5 + 25 = 30
their difference (25 - 5) = 20
1.5 of that = 30 , check!

if x = 1/5
sum of squares = 1/5 + 1/25 = 6/25
1.5(1/5 - 1/25) = 6/25 , check!

x = 0, 5, 1/5