To find the number, let’s represent it with a variable. We can use x to represent the unknown number.
According to the problem, the square of the number can be expressed as x2. The phrase ‘7 times the number’ can be written as 7x. Since the square of the number is 12 less than 7 times the number, we can set up the following equation:
x2 = 7x – 12
Now, let’s rearrange the equation to bring all terms to one side:
x2 – 7x + 12 = 0
Next, we can factor this quadratic equation. We’re looking for two numbers that multiply to 12 and add to -7. These numbers are -3 and -4.
So we can factor the equation as:
(x – 3)(x – 4) = 0
This gives us two possible solutions:
- x – 3 = 0 → x = 3
- x – 4 = 0 → x = 4
Thus, the two possible numbers are 3 and 4.
To verify our solutions, we can plug them back into the original condition:
- For x = 3:
32 = 9, and 7(3) – 12 = 21 – 12 = 9. So this solution works! - For x = 4:
42 = 16, and 7(4) – 12 = 28 – 12 = 16. So this solution works as well!
In conclusion, the numbers that satisfy the condition given in the problem are 3 and 4.