Question

(x+11+ 6(x+2)1/2)1/2+((x+11-6(x+2)1/2)1/2=x+2

Answer 1

(х+11+6х+12)1/2)1/2+((х+11-6х-12)1/2)1/2=х+2
((7х+23)1/2)/2+((-5х-1)1/2)1/2=х+2
(7х+23)1/2+(-5х-1)1/2=2х+4
7х+23-5х-1=4х+8
2х+22=4х+8
22-8=4х-2х
2х=14
х=7

Answer 2

Let's solve the equation step by step:

Given equation: √(x + 11 + 6√(x + 2)) + √(x + 11 - 6√(x + 2)) = x + 2

First, let's simplify the equation by isolating one of the square root terms on the left side:

√(x + 11 + 6√(x + 2)) = x + 2 - √(x + 11 - 6√(x + 2))

Now, square both sides of the equation to eliminate the square roots:

(x + 11 + 6√(x + 2)) = (x + 2 - √(x + 11 - 6√(x + 2)))^2

Expand the right side:

(x + 11 + 6√(x + 2)) = (x + 2 - √(x + 11 - 6√(x + 2))) * (x + 2 - √(x + 11 - 6√(x + 2)))

Now, let's simplify the equation step by step.

Let A = x + 11 and B = √(x + 2):

A + 6B = (x + 2 - √(A - 6B)) * (x + 2 - √(A - 6B))

Expand the right side:

A + 6B = (x^2 + 4x + 4 - 2x√(A - 6B) - 2x√(A - 6B) + A - 6B)

Now, simplify further:

A + 6B = (x^2 + 4x + 4 + A - 6B - 4x√(A - 6B))

Now, move terms around:

2x√(A - 6B) = 2B

Divide both sides by 2:

x√(A - 6B) = B

Now, square both sides again to eliminate the square root:

x^2(A - 6B) = B^2

Substitute A and B back in:

(x + 11)(x + 2 - 6√(x + 2)) = (x + 2)^2

Now, let's simplify and solve for x:

Expand the left side:

(x^2 + 13x + 22 - 6x√(x + 2)) = (x + 2)^2

Expand the right side:

x^2 + 13x + 22 - 6x√(x + 2) = x^2 + 4x + 4

Now, subtract x^2 from both sides to cancel out the x^2 terms:

13x + 22 - 6x√(x + 2) = 4x + 4

Subtract 4x from both sides:

9x + 22 - 6x√(x + 2) = 4

Subtract 22 from both sides:

9x - 6x√(x + 2) = -18

Divide both sides by 3 to simplify:

3x - 2x√(x + 2) = -6

Now, isolate the square root term:

2x√(x + 2) = 3x + 6

Divide both sides by 2x:

√(x + 2) = (3x + 6)/(2x)

Square both sides again:

x + 2 = (3x + 6)^2/(4x^2)

Now, expand the right side:

x + 2 = (9x^2 + 36x + 36)/(4x^2)

Multiply both sides by 4x^2 to eliminate the fractions:

4x^2(x + 2) = 9x^2 + 36x + 36

Now, expand and simplify:

4x^3 + 8x^2 = 9x^2 + 36x + 36

Subtract 9x^2 from both sides:

4x^3 - x^2 = 36x + 36

Now, subtract 36x and 36 from both sides:

4x^3 - x^2 - 36x - 36 = 0

This is a cubic equation. Solving cubic equations analytically can be quite complex, and there may not be a simple algebraic solution. You may need to use numerical methods or software to find approximate solutions for x.