Question

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

Answer 1

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

(x+11+6(x+2)1/2)1/2+((x+11-6(x+2)1/2)1/2=x+2
(x+11+3x+6)1/2+(x+11-3x-6)1/2=x+2умножаем на 2
x+11+3x+6+x+11-3x-6=2x+4
2х+22=2х+4
переносим все не известные в левую часть, а известные вправую
2х-2х=-22+4
приводим подобные слагаемые
0х=-18
нет решений

Answer 3

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 isolate one of the square roots on the left side of the equation:

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

Now, square both sides to eliminate the square root:

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

Expand the right side:

x + 11 + 6√(x + 2) = x^2 + 4x + 4 - 2(x + 2)√(x + 11 - 6√(x + 2)) + (x + 11 - 6√(x + 2))

Now, let's isolate the radical term (6√(x + 2)):

6√(x + 2) + 2(x + 2)√(x + 11 - 6√(x + 2)) = x^2 + 4x + 4 - (x + 11)

Next, let's move the terms involving the radical to the left side and the other terms to the right side:

6√(x + 2)√(x + 11 - 6√(x + 2)) = x^2 + 4x + 4 - (x + 11) - 2(x + 2)√(x + 11 - 6√(x + 2))

Now, we can simplify the right side:

6√(x + 2)√(x + 11 - 6√(x + 2)) = x^2 + 4x - 7 - 2(x + 2)√(x + 11 - 6√(x + 2))

Now, let's isolate the radical on the left side by dividing both sides by 6:

√(x + 2)√(x + 11 - 6√(x + 2)) = (x^2 + 4x - 7 - 2(x + 2)√(x + 11 - 6√(x + 2))) / 6

Square both sides again to eliminate the radicals:

x + 2(x + 11 - 6√(x + 2)) = (x^2 + 4x - 7 - 2(x + 2)√(x + 11 - 6√(x + 2))) / 36

Expand both sides:

x + 2x + 22 - 12√(x + 2) = (x^2 + 4x - 7 - 2(x + 2)√(x + 11 - 6√(x + 2))) / 36

Combine like terms:

3x + 22 - 12√(x + 2) = (x^2 + 4x - 7 - 2(x + 2)√(x + 11 - 6√(x + 2))) / 36

Now, let's get rid of the fraction by multiplying both sides by 36:

36(3x + 22 - 12√(x + 2)) = x^2 + 4x - 7 - 2(x + 2)√(x + 11 - 6√(x + 2))

Expand both sides:

108x + 792 - 432√(x + 2) = x^2 + 4x - 7 - 2(x + 2)√(x + 11 - 6√(x + 2))

Move all terms to one side of the equation:

x^2 - 104x + 799 + 430√(x + 2) - 2(x + 2)√(x + 11 - 6√(x + 2)) = 0

This is a quadratic equation in terms of √(x + 2) and requires further simplification. Unfortunately, it doesn't seem to have a simple analytical solution. Solving for x would likely require numerical methods or a computer algebra system to approximate a solution.