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.