How do you solve #sqrt(8-x)=x+6# and identify any restrictions?

Answer 1

The right side cannot be negative and neither can the argument of the radical.
Square both sides.
Solve the quadratic.
Discard any roots that violate the restrictions.

The restrictions are that the right side cannot be negative and the same for the argument of the radical:

#x+6 >= 0# and #8-x >=0#
#x >= -6# and #8 >=x#

Combining them:

#-6 <= x <= 8#

Add this restriction to the given equation:

#sqrt(8-x)=x+6; -6 <= x <= 8#

Square both sides:

#8-x = x^2+ 12x+ 36; -6 <= x <= 8#

Combine like terms:

#0 = x^2+ 13x+ 28; -6 <= x <= 8#

Use the quadratic formula:

#x = (-13+-sqrt(13^2-4(1)(28)))/(2(1)); -6 <= x <= 8#
#x = (-13+-sqrt(57))/2; -6 <= x <= 8#

The negative root us outside the restriction, therefore, we discard it:

#x = (-13+sqrt(57))/2#
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Answer 2

To solve the equation sqrt(8-x) = x+6, we can follow these steps:

  1. Square both sides of the equation to eliminate the square root: (sqrt(8-x))^2 = (x+6)^2.
  2. Simplify the equation: 8-x = x^2 + 12x + 36.
  3. Rearrange the equation to form a quadratic equation: x^2 + 13x + 28 = 0.
  4. Factorize the quadratic equation: (x+4)(x+7) = 0.
  5. Set each factor equal to zero and solve for x: x+4 = 0 or x+7 = 0.
  6. Solve for x in each equation: x = -4 or x = -7.

The restrictions for this equation arise from the square root function. Since the square root of a number cannot be negative, the expression inside the square root (8-x) must be greater than or equal to zero. Therefore, we have the restriction 8-x ≥ 0. Solving this inequality, we find x ≤ 8.

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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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