# Solve for x : sqrt(x+3) + sqrt(x-2) =4

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We have to solve sqrt(x+3) + sqrt(x-2) =4 and determine x.

First we square both the sides of the equation.

=> [sqrt(x+3) + sqrt(x-2)]^2 = 4^2

=> [sqrt(x+3) + sqrt(x-2)]^2 = 16

=> [sqrt(x+3)]^2 +[sqrt(x-2)]^2 + 2*[sqrt(x+3)]*[sqrt(x-2)] = 16

=> x+3 + x-2 +2*[sqrt(x+3)]*[sqrt(x-2)] = 16

=> 2x +1 + 2*[sqrt(x+3)]*[sqrt(x-2)] = 16

=> 2*[sqrt(x+3)]*[sqrt(x-2)] = 15 – 2x

Square both the sides again

=> 4*(x+3)(x-2) = 15^2 + 4x^2 – 60x

=> 4(x^2 + x -6) = 225 + 4x^2 – 60x

=> 4x^2 + 4x – 24 = 225 + 4x^2 – 60x

=> 64x= 249

=> x = 249 / 64

**Therefore x = 249 / 64**

To solve sqrt(x+3)+sqrt(x-2) = 8.

To solve this equation we square both sisides:

x+3 +2sqrt{(x+3)(x-2)} +x-2 = 4^2

2sqrt{x+3)(x-2)} + 2x+1 = 16

2sqrt((x+3)(x-2)} = 16 -(2x+1)

2sqrt(x+3)(x-2) = 15-2x.

We again square quare both sides:

4(x+3)(x-2) = (15-2x)^2

4{x^2-2x+3x-6} = 15^2-2*15*2x+4x^2

4x^2+4x-24 = 3969-60x+4x^2

Subtract 4x^2-60x-24 from both sides and rearrange:

(60+4)x = 225+24

64x = 249

x = 249/64

x = 3.890625.