The square of 9 less than a number is 3 less than the number. What is the number?

To solve this problem, let’s define the unknown number as x.

The phrase “the square of 9 less than a number” can be translated mathematically as (x – 9)². The phrase “3 less than the number” translates to x – 3.

So, we can set up the equation:

(x – 9)² = x – 3

Now, let’s expand the left side:

(x – 9)(x – 9) = x² – 18x + 81

This gives us:

x² – 18x + 81 = x – 3

Next, we want to move all terms to one side of the equation:

x² – 18x + 81 – x + 3 = 0

This simplifies to:

x² – 19x + 84 = 0

Now, we can factor the quadratic equation:

(x – 12)(x – 7) = 0

This gives us two potential solutions for x:

x = 12 or x = 7

To ensure both solutions are valid, we can verify them by substituting back into the original problem:

• For x = 12:
  The left side becomes (12 – 9)² = 3² = 9, and the right side becomes 12 – 3 = 9. Both sides are equal.
• For x = 7:
  The left side becomes (7 – 9)² = (-2)² = 4, and the right side becomes 7 – 3 = 4. Both sides are equal.

Thus, the solutions to the equation are 12 and 7.