How do you solve #x+2-2sqrt(x+3)=0# and find any extraneous solutions?

Answer 1
#x+2-2sqrt(x+3)=0#
#x+2=2sqrt(x+3)#

Raise to the square, keeping in mind that this will introduce erroneous answers:

#(x+2)^2=(2sqrt(x+3))^2#
#x^2+4x+4=4(x+3)#
#x^2+4x+4=4x+12#
#x^2+4=+12# #x^2-8=0# #x=+-sqrt(8)#

Now check the answers in order to rule out any incorrect answers:

only #+sqrt(8)# is correct.
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Answer 2

To solve the equation x+2-2sqrt(x+3)=0, we can follow these steps:

  1. Start by isolating the square root term by subtracting 2 from both sides of the equation: x - 2sqrt(x+3) = -2

  2. Next, move the square root term to one side of the equation by adding 2sqrt(x+3) to both sides: x = 2sqrt(x+3) - 2

  3. Square both sides of the equation to eliminate the square root: x^2 = (2sqrt(x+3) - 2)^2

  4. Expand the right side of the equation: x^2 = 4(x+3) - 8sqrt(x+3) + 4

  5. Simplify the equation further: x^2 = 4x + 12 - 8sqrt(x+3) + 4

  6. Rearrange the terms to isolate the square root term: x^2 - 4x - 16 = -8sqrt(x+3)

  7. Divide both sides of the equation by -8 to isolate the square root term: (x^2 - 4x - 16)/-8 = sqrt(x+3)

  8. Square both sides of the equation again to eliminate the square root: [(x^2 - 4x - 16)/-8]^2 = x + 3

  9. Simplify the equation further: (x^2 - 4x - 16)^2/64 = x + 3

  10. Expand and simplify the equation: (x^4 - 8x^3 + 32x^2 + 64x + 256)/64 = x + 3

  11. Multiply both sides of the equation by 64 to eliminate the fraction: x^4 - 8x^3 + 32x^2 + 64x + 256 = 64x + 192

  12. Rearrange the terms to form a quadratic equation: x^4 - 8x^3 + 32x^2 - 64x + 64 = 0

  13. Solve this equation using numerical methods or a graphing calculator to find the values of x.

To check for extraneous solutions, substitute each potential solution back into the original equation and verify if it satisfies the equation. If any solution does not satisfy the original equation, it is considered an extraneous solution.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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