Completing the Square Word problems - Help?

Question

A student is asked to do the following: "Choose any number. Subtract ten times the original number and subtract the square of the original number from 375. Find the greatest result and the value of the original number that gives the results."

a) If x is the original number a y is the result, write an equation that represents the instructions

b) Find the greatest result and the value of the original number that gives the greatest result

Please explain how you came to the answer :) Thanks!

Answer 1

a)
If I've read it correctly:
y = 375 - 10x - x²

b) Now it is time to complete the square. In general, if you have y = ax² + bx + c, completing the square will get you to a form of y = a(x - h) + k

Start by putting the terms in order x², x, 1
y = -x² - 10x + 375

First divide both sides by a (-1) to get x² by itself:
-y = x² + 10x - 375

Now complete the square by taking the x-term (10), dividing it in half (5) and squaring it (25). Add and subtract this from the right-hand side:

-y = x² + 10x + 25 - 25 - 375

If you group the first 3 terms, you have a perfect square:
-y = (x² + 10x + 25) - 25 - 375

So rewrite the part in parentheses as a square:
-y = (x + 5)² - 25 - 375

Combine the numbers to the right:
-y = (x + 5)² - 400

Finally divide by a (-1) to get the final form:
y = -(x + 5)² + 400

You know a square is non-negative. Anything bigger than zero, will subtract something from 400. So you want the square to be *exactly* zero.
x + 5 = 0
x = -5

If the number is -5, then the square will be 25 and ten times the number will be -50. Subtract these both from 375 and you get 400.

The greatest result is 400 and the original number is -5

Answer 2

this is fun mathematics!
give it a try yourself.
x = original number

y = -10x - (375-x)²

now you can do the fun part .. start trying different values of x and see how y changes

[HINT: try the x values between -6 and +3]