The only solution to sqrt (2x+3) - sqrt(x+1) = 1 is x = 3
I think that is true but I'm not sure if I was suppose to separate this equation I just put the 3 in the place of the x and it was true but should I have done it differently?
The answer is 3 and -1, from Wolframealpha
I arrived at the - 1 answer, but not sure if the way I solved it is correct and just luck that I got the correct answer of -1!!
maybe a tutor will step in an answer.
x = 3 does not satisfy the equation as a solution.
LS = (6+3) - √4
= 9-2
= 7
RS = 1
LS ≠ RS, so x ≠ 3
let's solve it ...
(2x+3) - sqrt(x+1) = 1
2x + 3 - 1 = √(x+1)
2x+2 = √(x+1)
square both sides
4x^2 + 8x + 4 = x+1
4x^2 + 7x + 3 = 0
(x+1)(4x+3) = 0
x = -1 or x = -3/4
if x = -1
LS = (-2+3) - √0
= 1-0 = 0
= RS
if x = -3/4
LS = 2(-3/4) + 3 - √(1/4)
= 3/2 - 1/2 = 1
= RS
so x = 0 or x = -3/4
Forget about my solution above,
Just plain ol' did not see that first square root sign!
√(2x+3) = √(x+1) + 1
square both sides
2x+3 = x+1 + 2√(x+1) + 1
x + 1 = 2√(x+1)
square again ...
x^2 + 2x + 1 = 4(x+1)
x^2 - 2x - 3 = 0
(x+1)(x-3) = 0
x = -1 or x = 3
checking both,since we squared.
if x = -1
LS = √(-2+3) - √0
= √1-√0 = 1 = RS
if x = 3
LS = √9 - √4
= 3-2
= 1 = RS
So x = -1 or x = 3
To solve the equation √(2x+3) - √(x+1) = 1, you did the right thing by substituting x = 3 and checking if it satisfies the equation. However, to ensure that it is the only solution, we need to demonstrate it more rigorously.
To solve the equation algebraically, we can follow these steps:
1. Start by isolating one of the radicals. For example, let's isolate √(2x+3):
√(2x+3) = 1 + √(x+1)
2. Square both sides of the equation to eliminate the radicals:
(√(2x+3))^2 = (1 + √(x+1))^2
2x + 3 = 1 + 2√(x+1) + x + 1
3. Simplify and collect like terms:
x + 3 = 2 + 2√(x+1) + x
4. Combine like terms and isolate the radical term:
2x - x + x - 2√(x+1) = 2 - 3
x - 2√(x+1) = -1
5. Now, square both sides again to eliminate the remaining radical:
(x - 2√(x+1))^2 = (-1)^2
x^2 - 4x√(x+1) + 4(x+1) = 1
6. Simplify and rearrange the equation:
x^2 - 4x√(x+1) + 4x + 4 = 1
x^2 - 4x√(x+1) + 4x + 3 = 0
7. At this point, you could attempt to factor or use the quadratic formula to solve for x. However, this equation is not a simple quadratic, and you may not be able to find an exact solution algebraically.
8. Therefore, you can use a numerical method to approximate the solution or use a graphing calculator to visualize the equation and find the approximate solution. One popular numerical method is the Newton-Raphson method.
However, if you have already tested x = 3 and found that it satisfies the equation, it is a strong indication that x = 3 is indeed the only solution.