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New Member
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Nov 20, 2007, 10:08 AM
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Algebra II solving with quadratic functions
Ok so I have failed to complete any of the word problems given to me in my Algebra II book, because I can not figure out how to manipulate the equations to get the correct information.
First problem I sort of figured but I'm still having trouble.
1. Find two numbers whose product is 120 and whose sum is a minimum.
I got - -
(x) * (120/x) = 120 so Sum(x) = x + 120/x which is right but how do I figure out whether it is a minimum or a maximum?? Do I put it into the formula y = a(x - h)^2 + k?? And if so how?
I'm trying to use the information from this problem to solve the next one which is NOT going well each time I come up against difficulties.
2. Find two numbers who difference is 8 and who product is a minimum. (let x = the first number and x + 8 = the second number)
Help me please I've spent 3 useless days on this problem, and now my life seems meaningless.
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Junior Member
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Nov 20, 2007, 10:34 AM
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Do you have any constraints like your results must be integers or something like that ?
Are you allowed to use calculus ? If you are, then differentiate the expressions you have figured out , look for a minima. Hope it helps.
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Junior Member
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Nov 20, 2007, 05:23 PM
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1. Let numbers be x, 120/x
Sum = y = x + 120/x
For minimum, dy/dx = 0, d^2y/dx^2 is positive.
dy/dx = 1 - 120/x^2
= 0 if x^2 = 120
x = sqrt (120)
d^2y/dx^2 = 240/x^3
At x = sqrt(120), d^2y/dx^2 is positive.
Hence x = sqrt (120) gives a minimum.
The other number = 120/x = sqrt (120)
Numbers are sqrt (120), sqrt (120)
Do similar procedure for second problem.
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Uber Member
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Nov 20, 2007, 10:05 PM
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I think you guys blew Ray away. Algebra II comes before Calculus. You might be scareing the poor person.
It does sound like a Calculus problem to me too.
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Junior Member
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Nov 21, 2007, 01:47 AM
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really, I am sorry then. I guessed so, that's why asked whether you are allowed to use calculus or not. Anyway, here is a non-calculus solution for you second question.
say one of the numbers is x and the other is x+8. Then their product p will be,
p=x*(x+8)
= x^2 + 8x
= x^2 + 2*x*4 + 4^2 - 4^2
= ( x + 4 )^2 - 4^2
-4^2 here is a constant. So we have to rely on ( x + 4 )^2 for the minimization task at hand. Since this is a squared term, the minimum value it can obtain is 0. From this reasoning we can say,
( x + 4 )^2 = 0
or, x + 4 = 0
or, x = -4
using this result you can find the other number, x+8 = 4.
So the numbers you were looking for are 4 and -4.
Hope this helps you :)
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Uber Member
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Nov 21, 2007, 08:41 AM
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Red:
The question stated "in my Algebra II book". Implicitly, that means no Calculus.
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Junior Member
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Nov 21, 2007, 08:47 AM
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Yeah, you are right. I should have understood that. But since I am not familiar with 'Algebra II' book, I decided to take a chance. In what class/standard/stage do they teach Algebra II ? High school ?
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New Member
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Nov 21, 2007, 08:00 PM
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Originally Posted by rayyrayy
Ok so I have failed to complete any of the word problems given to me in my Algebra II book, because I can not figure out how to manipulate the equations to get the correct information.
First problem I sort of figured but I'm still having trouble.
1. Find two numbers whose product is 120 and whose sum is a minimum.
I got - -
(x) * (120/x) = 120 so Sum(x) = x + 120/x which is right but how do I figure out whether it is a minimum or a maximum??? Do I put it into the formula y = a(x - h)^2 + k ??? and if so how?
I'm trying to use the information from this problem to solve the next one which is NOT going well each time I come up against difficulties.
2. Find two numbers who difference is 8 and who product is a minimum. (let x = the first number and x + 8 = the second number)
Help me please I've spent 3 useless days on this problem, and now my life seems meaningless.
The numbers are x and 120-x, so those are the number to put into:
x*(120-x)=120.
For a good explanation of how to solve this or any quadratic equation, go to:
Free Online Quadratic Equation Solver: Solve by Most Efficient Method
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Uber Member
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Nov 21, 2007, 08:29 PM
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A long, long, long time ago, I remember algebra, algebra I, algebra II, Geometry, Pre-calc, Calculus, differential equations. With logic, statistics and probability thrown in someplace.
There is overlap, but it definitely progressed in that order.
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