Question

Sqrt(x+6-4*sqrt(x+2))+sqrt(11+x-6*sqrt(x+2))=1

Answer 1

Ответ: x=7.

Пошаговое объяснение:

√(x+6-4*√(x+2))+√(11+x-6*√(x+2))=1

√((x+2)-2*2*√(x+2)+4)+√((x+2)-2*3*√(x+2)+9)=1

√((x+2)-2*2*√(x+2)+2²)+√((x+2)-2*3*√(x+2)+3²)=1

√(√(x+2)-2)²+√(√(x+2)-3)²=1

|√(x+2)-2|+|√(x+2)-3|=1

√(x+2)-2+√(x+2)-3=1

2*√(x+2)-5=1

2*√(x+2)=6  |÷2

√(x+2)=3

(√(x+2))²=3²

x+2=9

x=7.

Answer 2

To solve the given equation:

√(x+6-4√(x+2)) + √(11+x-6√(x+2)) = 1

Let's simplify the equation step by step.

1. Start by simplifying the expressions inside the square roots. √(x+6-4√(x+2)) can be written as √[(√(x+2))^2 - 2*√(x+2)*2 + 2^2]. Simplifying further, we get √[(√(x+2) - 2)^2].

   Similarly, √(11+x-6√(x+2)) can be written as √[(√(x+2))^2 - 2√(x+2)(-6) + (-6)^2]. Simplifying further, we get √[(√(x+2) - 6)^2].

2. Now our equation becomes: √[(√(x+2) - 2)^2] + √[(√(x+2) - 6)^2] = 1

3. Since the sum of two square roots is equal to 1, we can write: (√(x+2) - 2) + (√(x+2) - 6) = 1

4. Simplify further: √(x+2) - 2 + √(x+2) - 6 = 1

5. Combine like terms: 2√(x+2) - 8 = 1

6. Move the constant term to the other side: 2√(x+2) = 1 + 8 2√(x+2) = 9

7. Square both sides of the equation to eliminate the square root: (2√(x+2))^2 = 9^2 4(x+2) = 81

8. Expand and simplify: 4x + 8 = 81

9. Move the constant term to the other side: 4x = 81 - 8 4x = 73

10. Divide both sides by 4 to solve for x: x = 73/4

Therefore, the solution to the given equation is x = 73/4 or 18.25.