Question

The sum of a number and it's square is one and a half times their difference. What is the number?

Answer 1

let the number be x,its square is ${x}^{2}$
$x + {x}^{2} = \frac{3}{2} (x - {x}^{2} ) \\ 2x + 2 {x}^{2} = 3x - 3 {x}^{2} \\ 5 {x}^{2} - x = 0 \\ x(5x - 1) = 0 \\ x = 0 \\ 5x - 1 = 0 \\ x = \frac{1}{5}$
$\\ \frac{1}{5} + \frac{1}{25} = \frac{5 + 1}{25} = \frac{6}{25} \\ \\ \frac{3}{2} ( \frac{1}{5} - \frac{1}{25} ) = \frac{3}{2} ( \frac{5 - 1}{25} ) \\ = \frac{3 \times 4}{2 \times 25} \\ = \frac{6}{25} \\ hence \: proved$
so that number is 1/5. answer