SOLUTION: Among all pairs of numbers (x,y) such that 3x+y=15, find the pair for which the sum of squares, x^2+y^2, is minimum. Write your answers as fractions reduced to lowest terms.

Algebra.Com
Question 1150050: Among all pairs of numbers (x,y) such that 3x+y=15, find the pair for which the sum of squares, x^2+y^2, is minimum. Write your answers as fractions reduced to lowest terms.
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.

y = 15-3x


Hence,  x^2 + y^2 = x^2 + (15-3x)^2 = x^2 + 225 - 90x + 9x^2 = 10x^2 - 90x + 225.


A quadratic function ax^2 + bx + c   with positive coefficient  "a"  has the minimum at  x = .


In your case,  a = 10  and  b = -90,  hence  x = - = .


Then  y =  =  = .


ANSWER.  This pair is  (x,y) = (,).

Solved.

------------------

On finding the maximum/minimum of a quadratic function,  see the lessons
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.


RELATED QUESTIONS

among all pairs of numbers (x,y) such that 4x+y=11 find the pair for which the sum of... (answered by Fombitz)
Among all pairs of numbers (x,y) such that 2x+y=8, find the pair for which the sum of... (answered by Theo)
Among all pairs of numbers (x,y) such as 4x+y=17 find the pair of which the sum o squares (answered by ankor@dixie-net.com)
Among all pairs of numbers (x,y) such that 2x+y=16, find the pair for which the sum of... (answered by josgarithmetic)
Among all pairs of numbers (x,y) such that 4x+y=18, find the pair for which the sum of... (answered by jim_thompson5910,ikleyn)
Among all pairs of numbers (x,y) such that 8x+2y=18, find the pair for which the sum of... (answered by ankor@dixie-net.com,ikleyn)
Among all pairs of numbers (x,y) such that 4x+2y=24, find the pair for which the sum of... (answered by greenestamps)
Among all pairs of numbers (x,y) such that 6x+2y=38, find the pair for which the sum of... (answered by Edwin McCravy,ikleyn)
Among all pairs of numbers (x,y) such that 4x+2y=22, find the pair for which the sum of... (answered by ikleyn)