Question

Please help me!!!!! I need all the work explained and it would mean the world

Answer 1

Answer:

• x = -4 . . . actual solution
• x = -7 . . . extraneous solution

Step-by-step explanation:

I like to solve these using a graphing calculator. You can plot the left side and right side (as below) and look for points of intersection. Here, the point of intersection is at x = -4, so that is the only solution.

If we plot the negative branch of the square root function, we can see that an extraneous solution would be x = -7. We expect to get that solution as part of a conventional solution process for this equation.

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My preference for solving these graphically is to write the equation as ...

  (something) = 0

A graphing calculator is usually very good at finding zeros of a function to high precision, so that works well. Here, I might use ...

  f(x) = √(x +8) -6 -x

and look for x that makes f(x) = 0. (see second attachment)

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The usual way to solve these is to isolate the radical, then raise the equation to a power that eliminates the radical. Here, we can add 6 to get ...

  √(x +8) = x +6

  x +8 = (x +6)² . . . . . . . .square both sides of the equation

  x² +11x +28 = 0 . . . . . . eliminate parentheses, subtract (x+8)

  (x +7)(x +4) = 0 . . . . . . factor the equation

Solutions that make these factors zero are ...

  x = -7   and   x = -4

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Trying these in the original equation, we get ...

For x = -7

  √(-7+8) -6 = -7

  1 - 6 = -7 . . . . . . . NOT TRUE . . . extraneous solution

For x = -4

  √(-4+8) -6 = -4

  2 -6 = -4 . . . . . . . TRUE, actual solution

The solution is x = -4; an extraneous solution is x = -7.