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The equation x2+y2−16x+6y+73=100 can be rewritten as [g(x)]2+(y+3)2=100.Write an expression for g(x).

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facundo3141592

Answer: g(x) = (x - 8)

Step-by-step explanation:

We start with the equation:

x^2 + y^2 - 16*x + 6*y + 73 = 100.

And we want to write this as:

g(x)^2 + (y + 3)^2 = 100.

So both equations are equal, then we can write this as:

x^2 + y^2 - 16*x + 6*y + 73 = g(x)^2 + (y + 3)^2

First we can expand the right side:

g(x)^2 + (y + 3)^2 = g(x)^2 + y^2 + 2*3*y + 9

Then we have:

x^2 + y^2 - 16*x + 6*y + 73 = g(x)^2 + y^2 + 6*y + 9

Now we can cancel the equal terms in each side, like y^2 and 6*y

x^2 - 16*x + 73 = g(x)^2 + 9

Now let's subtract 9 in each side:

x^2 - 16*x + 73 - 9 = g(x)^2

x^2 - 16*x + 64 = g(x)^2

Then we can assume that g(x) = (x - a)

then:

x^2 - 16*x + 64 = (x - a)^2 = x^2 - 2*a*x + a^2

-16*x + 64 = -2*a*x + a^2

now we must find the value of a, we have two equations:

-2*a = -16

a^2 = 64

and each equation yields to:

a = -16/-12 = 8.

a = √64 = 8

Then we have: g(x) = (x - 8)

Completing the squares, it is found that: [tex]g(x) = x - 8[/tex]

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The expression given is:

[tex]x^2 + y^2 - 16x + 6y + 73 = 100[/tex]

To complete the squares:

  • First, each first order term(-16 and 6) are divided by 2.
  • Then, we write as the square of (x - 8) and (x + 3).
  • Then, we have to find the squares on the other side of the equality.

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Doing these steps:

[tex](x - 8)^2 + (y + 3)^2 = 100 - 73 + 8^2 + 3^2[/tex]

[tex](x - 8)^2 + (y + 3)^2 = 100 - 73 + 64 + 9[/tex]

[tex](x - 8)^2 + (y + 3)^2 = 100[/tex]

Which means that:

[tex]g(x) = x - 8[/tex]

A similar problem is given at https://brainly.com/question/16269892