Solve the equation: √5x+1-√2x+3=1?

√(5x + 1) - √(2x + 3) = 1

square both sides,
(5x + 1) - 2√[(5x + 1)(2x + 3)] + (2x + 3) = 1
(5x + 1) + (2x + 3) - 1 = 2√[(5x + 1)(2x + 3)]
2√[(5x + 1)(2x + 3)] = 7x + 3

square both sides again,
4(5x + 1)(2x + 3) = (7x + 3)^2
4(10x^2 + 17x + 3) = 49x^2 + 42x + 9
40x^2 + 68x + 12 = 49x^2 + 42x + 9
9x^2 - 26x - 3 = 0
(9x + 1)(x - 3) = 0
x = -1/9 or x = 3

but we have to remember that
5x + 1 > 2x + 3 ≥ 0

5x + 1 > 2x + 3
3x > 2
x > 2/3 , AND
2x + 3 ≥ 0
x ≥ -3/2

so our limit is x > 2/3

answer : x = 3

[√(3x-2)]^2 = 1^2
3x-2=1
3x=3
x=1

√(5x+1) - √(2x+3) = 1
√(5x+1) = 1 + √(2x+3)

square both sides
5x+1 = 1 + 2√(2x+3) + 2x+3
3x - 3 = 2√(2x+3)

square both sides
9x² - 18x + 9 = 4(2x+3)
9x² - 18x + 9 = 8x+12
9x² - 10x - 3 = 0

Use the quadratic formula to solve
   ax²+bx+c=0
where
   a=9
   b=-10
   c=-3

x = [-b ± √(b²-4ac)] / (2a)
   = [-(-10) ± √((-10)² - 4(9)(-3))] / (2·9)
   = [10 ± √(208)] / 18
   = [10 ± √(2⁴)√(13)] / 18
   = [10 ± 4√13] / 18
   ≅ 0.556 ± 0.801

x₁ = 0.556 - 0.801 = -0.246
x₂ = 0.556 + 0.801 = 1.357

√(5x + 1) - √(2x + 3) = 1
√(5x + 1) = 1 + √(2x + 3)
5x + 1 = [1 + √(2x + 3)]^2
5x + 1 = [1 + √(2x + 3)][1 + √(2x + 3)]
5x + 1 = 1*1 + 1*√(2x + 3) + √(2x + 3)*1 + √[(2x + 3)(2x + 3)]
5x + 1 = 1 + √(2x + 3) + √(2x + 3) + 2x + 3
5x + 1 = 1 + 2√(2x + 3) + 2x + 3
5x + 1 = 2x + 1 + 3 + 2√(2x + 3)
5x + 1 = 2x + 4 + 2√(2x + 3)
5x + 1 - 2x - 4 = 2√(2x + 3)
(3x - 3)/2 = √(2x + 3)
[(3x - 3)/2]^2 = 2x + 3
[(3x - 3)(3x - 3)/(2^2)] = 2x + 3
(9x^2 - 9x - 9x + 9)/4 = 2x + 3
9x^2 - 18x + 9 = 4(2x + 3)
9x^2 - 18x + 9 = 8x + 12
9x^2 - 18x - 8x + 9 - 12 = 0
9x^2 - 26x - 3 = 0
9x^2 + x - 27x - 3 = 0
(9x^2 + x) - (27x + 3) = 0
x(9x + 1) - 3(9x + 1) = 0
(9x + 1)(x - 3) = 0

9x + 1 = 0
9x = -1
x = -1/9 (extraneous solution)

x - 3 = 0
x = 3

∴ x = 3

Parentheses are your friend. Is this sqrt(5x) + 1 or sqrt(5x + 1)?

I'm going to assume it's sqrt(5x + 1) - sqrt(2x + 3) = 1

Square both sides

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

(5x + 1) - 2 * sqrt(5x + 1)sqrt(2x + 3) + (2x + 3) = 1

7x + 4 - 2 * sqrt ((5x + 1)(2x + 3)) = 1

7x + 3 -2 * sqrt(10x^2 + 17x + 3) = 0

2 * sqrt(10x^2 + 17x + 3) = 7x + 3

Square both sides again

4 * (10x^2 + 17x + 3) = 49x^2 + 42x + 9

40x^2 + 68x + 12 = 49x^2 + 42x + 9

9x^2 - 26x - 3 = 0

x = (26 +/- sqrt((-26)^2 - 4 * 9 * -3)) / (2 * 9)
x = (26 +/- sqrt(676 + 108)) / 18
x = (26 +/- sqrt(784)) / 18
x = (26 +/- 28) / 18

x = -2 / 18 or x = 54 / 18

x = -1/9 or x = 3

We did a lot of squaring in the solving, and one of the problems with that is that sometime it can generate errors of sign. Because sqrt(4) = 2, but 4^(1/2) = 2 or -2. So we have to test our answers by plugging them back into the original formula:

sqrt(5*3 + 1) - sqrt(2*3 + 3) = 1
sqrt(15 + 1) - sqrt(6 + 3) = 1
sqrt(16) - sqrt(9) = 1
4 - 3 = 1

sqrt(5*-1/9 + 1) - sqrt(2*-1/9 + 3) = 1
sqrt(-5/9 + 1) - sqrt(-2/9 + 3) = 1
sqrt(4/9) - sqrt(25/9) = 1
2/3 - 5/3 = -1 < See, error in sign.

So the answer is 3

the solution is x=3. but then the square root is over 5x+1 as well as 2x+3. only then x=3.if you have any further problem go to mathnerds.com you will get a satisfactory answer.

[√5(x + 1) – √(2x + 3) = 1
[√5(x + 1) = 1 + √(2x + 3) squaring both sides
5(x + 1) = 1 + 2√(2x + 3) + 2x + 3
5x + 5 = 4 + 2√(2x + 3) + 2x
3x + 1 = 2√(2x + 3) squaring both sides
9x² + 6x + 1 = 4(2x + 3)
9x² + 6x + 1 = 8x + 12
9x² – 2x – 11 = 0
9x² + 9x – 11x – 11 = 0
9x(x + 1) – 11(x + 1) = 0
(9x – 11)(x + 1) = 0
x = 11/9 or x = – 1
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Please clarify where your square roots end. Is it

sqrt(5)*x+1-sqrt(2)*x+3=1

or perhaps

sqrt(5x+1)+sqrt(2x+3)=1

or something else?

That second one seems the most likely to me, but it's not clear from your description.

I assume that this is:

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

Square both sides of this equation, getting:

5x + 1 - (2*(sqrt(5x + 1) * sqrt(2x + 3)) + 2x + 3 = 1

7x + 4 + (2*(sqrt(5x + 1) * sqrt(2x + 3)) = 1

Note: sqrt(a) * sqrt(b) = sqrt(a * b)....................So the above becomes:

7x + 4 + (2 * sqrt(10x^2 + 17x + 3)) = 1.....Subtract 1 from both sides.

7x + 3 + (2 * sqrt(10x^2 + 17x + 3)) = 0

Subtract (2 * sqrt(10x^2 + 17x + 3)) from both sides:

7x + 3 = -(2 * sqrt(10x^2 + 17x + 3))......Square both sides


(49X^2 + 42x + 9) = 4 * (10x^2 + 17x + 3)

49x^2 + 42x + 9 = 40x^2 + 68x + 12....Subtract 40x^2 + 68x + 12 from both sides:

9x^2 -26x - 3 = 0

Factor: Factoring 9 as 3 * 3 cannot possibly work so we must have:

(9x + 1) * (x - 3) = 0

x = 3 and x = (-1/9)

However when we run around squaring things we may get extra answers, so now we have to check both answers

3 -> sqrt(25) - sqrt(16) = 5 - 4 = 1.....Check

(-1/9)-> sqrt(4/9) - sqrt(25/9) = 2/3 - 5/3 = ***-1***. The squaring turns a -1 into a +1.

Answer: 3.....<<<<<.....This is the only Answer
_______________

I used a graphics program to check my result and 3 is the only answer. :)
.

i am confused this really looks simple but everyone made it so hard
√5x+1-√2x+3=1squaring both sides
5x+1-(2x+3)=1
5x+1-2x-3=1
3x+(-2)=1
3x=1+2
3x=3
x+1