anonymous
  • anonymous
Let f(X) be the square of the distance from the oint (2,1) to a point (x, 3x+2) on the line y=3x+2. Show that f(x) is a quadratic function, and find its minimum value by completing the square.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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ganeshie8
  • ganeshie8
f(x) = square of distance between points (2, 1) and (x, 3x+2)
ganeshie8
  • ganeshie8
use the distance formula and write down f(x)
anonymous
  • anonymous
is that it?

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ganeshie8
  • ganeshie8
wat did u get after applying distance formula ?
anonymous
  • anonymous
I got root (-10x^2 +13)
ganeshie8
  • ganeshie8
ok, lets see if its correct :)
ganeshie8
  • ganeshie8
we have the distance formula , distance between two points = \(\large \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
ganeshie8
  • ganeshie8
distance between points (2, 1) and (x, 3x+2) = \(\sqrt{(x-2)^2+(3x+2-1)^2}\) \(\sqrt{(x-2)^2+(3x+1)^2}\) \(\sqrt{x^2+4-4x+ 9x^2+1+6x}\) \(\sqrt{10x^2+2x+5}\)
ganeshie8
  • ganeshie8
f(x) = square of distance between points (2, 1) and (x, 3x+2)
ganeshie8
  • ganeshie8
so, f(x) = \((\sqrt{10x^2+2x+5})^2\) = \(10x^2+2x+5\)
ganeshie8
  • ganeshie8
thats our required function f(x), clearly, its a quadratic (why ?)
anonymous
  • anonymous
Oh I see, I foiled it wrong thank you so much!
ganeshie8
  • ganeshie8
np :) you still need to find its minimum value...
anonymous
  • anonymous
Its minimum value would be the lowest point on the graph c:
ganeshie8
  • ganeshie8
Yes, u knw how to find it ha ? :)
anonymous
  • anonymous
Can you review over how to for me?
ganeshie8
  • ganeshie8
ok sure :) we simply need to change the quadratic to form : \(\large a(x-h)^2+\color{red}{k}\) \(\color{red}{k}\) is our required minimum value
ganeshie8
  • ganeshie8
\(10x^2+2x+5\) factor out 10 \(10(x^2+2/10x+5/10)\)
anonymous
  • anonymous
oh okay thank you! I know now, 49/10
ganeshie8
  • ganeshie8
\(10x^2+2x+5\) factor out 10 \(10(x^2+2/10x+5/10)\) \(10(x^2+2(1/10)x+ 1/100 - 1/100 + 5/10)\) \(10((x+1/10)^2 - 1/100 + 5/10)\) \(10((x+1/10)^2 - 1/100 + 50/100)\) \(10((x+1/10)^2 + 49/100)\) \(10(x+1/10)^2 + 10 \times 49/100\) \(10(x+1/10)^2 + 49/10\)
ganeshie8
  • ganeshie8
Yes, \(\color{red}{k}\) = 49/10, so thats our min value !

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