# Solve for x sqrt(2x+5) = sqrt(x+2) + sqrt(2x-3).

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sqrt(2x+5) = sqrt(x+2) + sqrt(2x-3)

Let us square both sides:

==> (2x+5) = (x+2) + 2sqrt(x+2)(2x-3) + (2x-3)

==> (2x+5) = x+2 + 2x -3 + 2sqrt(2x^2 +x -6)

==> (2x+5) = 3x -1 + 2sqrt(2x^2 + x -6)

==> -x +6 = 2sqrt(x+2)(2x-3)

Square both sides again:

==> x^2 -12x + 36 = 4*(2x^2 + x - 6)

==> x^2 -12x +36 = 8x^2 +4x - 24

Group similar terms:

==> 7x^2 +16x -60 = 0

==> x1= [-16 + sqrt(25d + 4*7*60)]/14= (-16 + 44)14

= 28/14= 2

==> x2= (-16 -44)/14= -60/14= -30/7

==> the solutions are:

x1= 2 and x2= -30/7

Before solving the equation, we'll set the constraint of existence of the sqrt.

2x + 5 >=0

2x>=-5

x>=-5/2

x+2>=0

x>=-2

2x-3>=0

2x>=3

x>=3/2

The interval of admissible values for x is [3/2 , +inf.)

Now, we'll solve the equation, raising to square both sides, in order to eliminate the square root:

[sqrt(2x+5)]^2 = [sqrt(x+2) + sqrt(2x-3)]^2

2x + 5 = x + 2 + 2x - 3 + 2sqrt(x+2)(2x-3)

We'll keep only the sqrt to the right side, the rest of the terms we'll move them to the left side:

2x + 5 -x - 2 - 2x + 3 = 2sqrt(x+2)(2x-3)

We'll eliminate like terms:

6-x = 2sqrt(x+2)(2x-3)

We'll raise to square again:

(6-x)^2 = 4*(x+2)(2x-3)

We'll expand the square and we'll remove the brackets:

36 - 12x + x^2 = 8x^2 + 4x - 24

We'll move all terms to one side:

36 - 12x + x^2 - 8x^2 - 4x + 24 = 0

We'll combine like terms:

-7x^2 - 16x + 60 = 0

We'll multiply by -1:

7x^2 + 16x - 60 = 0

We'll apply the quadratic formula:

x1 = (-16 + 44) / 14

x1 = 2

x2 = (-16 - 44) / 14

x2 = -60 / 14

x2 = -30/7

**Since the value of the first root belongs to the interval of admissible values of x, the equation will have only one solution, namely x = 2.**

sqrt(2x+5) = sqrt(x+2)+sqrt(2x-3). To solve for x.

We squaring both sides we get

2x+5 = x+2 + 2sqrt{(x+2)(2x-3)} +2x-3

2x+5 = 3x-1 +2sqrt{x+2)(2x-3)}

2x+5-3x+1 = 2sqrt{(x+2)(2x-3)}

-x+6 = 2sqrt{x+2)(2x-3)}. Again sqre both sides:

(-x+6)^2 = 4(x+2)(2x-3)

x^2-24x+36= 4{2x^2 -3x+4x-6}

x^2-24x+36 = 8x^2 +4x-24

0 = (8-1)x^2 +(24+4)x -24-36

7x^2 +28x-60 = 0

x = {-28 +or- sqrt(28^2 - - 4*7*60}/2*7

x1 = {-28 + sqrt(2864)}/14.

x2 {-28 -sqrt(2864)}/14