Final answer:
To solve each of the given equations, we need to eliminate the square roots and isolate the variable x.
Step-by-step explanation:
Let's solve each equation step-by-step:
1. √(x-3)=√(x+15)-2
Square both sides to eliminate the square roots:
(x-3) = (x+15) - 2√(x+15) + (x+15)
x - 3 = x + 15 - 2√(x+15) + x + 15
-3 = 30 - 2√(x+15) + 2x
Combine like terms:
-3 = 30 + 2x - 2√(x+15)
Rearrange the equation:
2x - 2√(x+15) = -3 - 30
Combine like terms:
2x - 2√(x+15) = -33
Divide by 2 to isolate the square root:
x - √(x+15) = -16.5
Now square both sides again:
x^2 - 2x√(x+15) + (x+15) = 272.25
x^2 - 2x√(x+15) + x + 15 = 272.25
Combine like terms:
x^2 - x + 15 - 272.25 = 2x√(x+15)
x^2 - x - 257.25 = 2x√(x+15)
Now square both sides again:
(x^2 - x - 257.25)^2 = 4x^2(x+15)
Solve this expanded equation:
x^4 - 2x^3 + 259.25x^2 + 2x^3 - 4x^2 - 404.25x - 257.25x + 514.5 = 4x^3 + 60x^2
Simplify and combine like terms:
x^4 + x^3 - 348.5x^2 - 661.5x + 514.5 = 4x^3 + 60x^2
Move all terms to one side of the equation:
x^4 + x^3 - 408.5x^2 - 721.5x + 514.5 = 0
This is a quartic equation and it is difficult to solve without further simplification or using numerical methods.
2. √(x+16)=x-√(x+17)
Square both sides to eliminate the square roots:
(x+16) = x^2 - 2x√(x+17) + (x+17)
x + 16 = x^2 - 2x√(x+17) + x + 17
Combine like terms:
16 = 2x - 2√(x+17)
Isolate the square root by moving terms around:
2√(x+17) = 2x - 16
Square both sides again:
(2√(x+17))^2 = (2x - 16)^2
4(x+17) = 4x^2 - 64x + 256
Expand and simplify:
4x + 68 = 4x^2 - 64x + 256
Cross out the common terms:
68 = 4x^2 - 64x - 64x + 256
Simplify and combine like terms:
68 = 4x^2 - 128x + 256
Move all terms to one side of the equation:
4x^2 - 128x + 188 = 0
This is a quadratic equation and can be solved using factoring, the quadratic formula, or completing the square.
3. √(x-3)-√(x-2)=1
Square both sides to eliminate the square roots:
(x-3) - 2√(x-3)(x-2) + (x-2) = 1
Combine like terms:
2x - 5 - 2√(x-3)(x-2) = 1
Isolate the square root term:
2√(x-3)(x-2) = 2x - 6
Square both sides again:
4(x-3)(x-2) = (2x - 6)^2
Expand and simplify:
4(x^2 - 2x - 3x + 6) = 4(x^2 - 4x + 4)
4(x^2 - 5x + 6) = 4(x^2 - 4x + 4)
Cross out the common terms:
4(x^2 - 5x + 6) = 4(x^2 - 4x + 4)
Divide by 4:
x^2 - 5x + 6 = x^2 - 4x + 4
Subtract x^2 from both sides:
-5x + 6 = -4x + 4
Subtract 4 from both sides:
-5x + 2 = -4x
Add 5x to both sides:
2 = x
4. √(√(x-3)) = √(x-15)
Square both sides to eliminate the square roots:
(√(x-3))^2 = (√(x-15))^2
x - 3 = x - 15
Subtract x from both sides:
-3 = -15
This equation has no solution.