Question 909123: Among all pairs of numbers (x,y) such that 2x+y=8, 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 Theo(13342)   (Show Source):
You can put this solution on YOUR website! your equation is 2x + y = 8
solve for y to get y = 8 - 2x
in the expression x^2 + y^2, replace y with 8 - 2x to get:
x^2 + (8-2x)^2
simplify to get x^2 + 64 - 32x + 4x^2
combine like terms an reorder the terms to get:
5x^2 - 32x + 64
this is the expression you want minimize.
since it's a quadratic equation, you can use the formula for the min/max point to find the vertex of this equation.
since the equation is in standard form, you can find a,b,c to get:
a = 5
b = -32
c = 64
the x value of the min/max point is x = -b/2a which becomes x = 32/10 which becomes x = 3.2
when x = 3.2, 5x^2 - 32x + 64 becomes 5*3.2^2 - 32*3.2 + 64 which becomes 51.2 - 102.4 + 64 which becomes x = 12.8
since the coefficient of the x^2 term is positive, that becomes a minimum point which is the value that your are looking for.
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