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Solve each equation and answer

1. √(x-3)=√(x+15)-2
2. √(x+16)=x-√(x+17)
3. √(x-3)-√(x-2)=1
4. √(√(x-3)) =√(x-15)

1 Answer

6 votes

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.

User Bollo
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