Final answer:
The solution to the equation where 36 subtracted from the square of a number equals five times the number is 9, as the equation x² - 36 = 5x leads to the valid solution of x = 9 after factoring and checking the possible answers.
Step-by-step explanation:
The student's question asks when 36 is subtracted from the square of a number, the result is five times the number, and it is looking to solve for the value of the number. To solve this, we set up the equation x² - 36 = 5x, where x represents the number in question.
First, we move all terms to one side to get the standard form of a quadratic equation, which is ax² + bx + c = 0. Our equation becomes x² - 5x - 36 = 0. Next, we can either factor the quadratic, complete the square, or use the quadratic formula to find the value of x.
If factoring, we look for two numbers that multiply to -36 and add to -5. These numbers are -9 and 4. This gives us the factors (x - 9)(x + 4) = 0, leading to the possible solutions of x = 9 or x = -4. We have to check both solutions in the original equation to determine which, if any, is valid.
After substitution, we find that 9 is the correct solution because 9² - 36 equals 81 - 36, which is indeed 45, and 45 is five times 9. Therefore, the value of the number is 9.