The difference between a number and its square is 72. What is the number?

Step-by-Step Solution

To find the number, we can set up an algebraic equation based on the problem statement.

1. Define the variables:

   • Let the number be x.

   • The square of the number is x^2.

   • The difference between the number and its square is 72. This can be written as: x^2 - x = 72.

2. Solve the quadratic equation:

   • Rearrange the equation into standard form: x^2 - x - 72 = 0.

   • Factor the quadratic equation: We need two numbers that multiply to -72 and add to -1. These are -9 and 8.

   • (x - 9)(x + 8) = 0.

   • This gives two potential solutions for x: x = 9 or x = -8.

3. Check the solutions:

   • If x = 9: The square is 81. The difference 81 - 9 = 72. This works.

   • If x = -8: The square is 64. The difference 64 - (-8) = 64 + 8 = 72. This also works.

   • Based on the options provided in your image, 9 is the correct choice.