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

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

Let's assume the number is represented by 'x'.

According to the given information, the sum of the number and its square is equal to one and a half times their difference.

So, we can write this as an equation: x + x^2 = 1.5(x - x^2).

To solve this equation, let's simplify it:
x + x^2 = 1.5x - 1.5x^2.

Now, rearrange the equation:
x^2 + 1.5x^2 = 1.5x - x.

Combine like terms:
2.5x^2 = 0.5x.

Divide both sides of the equation by 0.5x:
2.5x^2 / 0.5x = x.

Simplify the left side of the equation:
5x = x.

Divide both sides of the equation by x:
5 = 1.

This means that there is no number that satisfies the given condition. The equation leads to a contradiction, indicating that there is no solution.

To solve this problem, let's break it down step by step.

Let's assume that the number is represented by 'x'. According to the problem, the sum of a number and its square is one and a half times their difference. We can represent this in an equation:

x + x^2 = 1.5 * (x - x^2)

Now, we need to solve this equation to find the value of 'x'.

Step 1: Expand
x + x^2 = 1.5x - 1.5x^2

Step 2: Bring all terms to one side
x + x^2 - 1.5x + 1.5x^2 = 0

Step 3: Combine like terms and simplify
2.5x^2 - 0.5x = 0

Step 4: Factor out common terms
x(2.5x - 0.5) = 0

Step 5: Solve for 'x' by setting each factor equal to zero
x = 0 or 2.5x - 0.5 = 0

For the first solution, x = 0.

For the second solution, solve the equation:
2.5x - 0.5 = 0
2.5x = 0.5
x = 0.5 / 2.5
x = 0.2

So, the possible values for 'x' are 0 and 0.2.