Question 1070500
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The sum of two numbers is 8. Find the numbers if the sum of their squares is a minimum. (hint completing the square)
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<pre>
Let x be one of the two numbers.
Then the other number is 8-x, and the sum of their squares is

  q(x) = {{{x^2 + (8-x)^2}}}     (1)

= {{{x^2 + 64 - 16x + x^2}}} = {{{2x^2 - 16x +64}}}.


This quadratic function has the minimum at x = {{{-b/(2a)}}} = {{{-(-16)/(2*2)}}} = {{{16/4}}} = 4.


Calculate this maximum value. For it, substitute the value x= 4 into the function (1):

q(4) = {{{4^2 + (8-4)^2}}} = 16 + 16 = 32.


<U>Answer</U>. The minimum is achieved at x = 4 and is equal to 16.

               The vertex of the quadratic function is (4,16).
</pre>


On finding the maximum/minimum of a quadratic function see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>HOW TO complete the square to find the minimum/maximum of a quadratic function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-How-to-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>Briefly on finding the minimum/maximum of a quadratic function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-to-find-the-vertex-of-a-quadratic-function.lesson>HOW TO complete the square to find the vertex of a parabola</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-finding-the-vertex-of-a-parabola.lesson>Briefly on finding the vertex of a parabola</A>



Also, you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


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