I find it curious here:
a - b=2 in this case,
rationalize by multplying both sides by a+b, getting
a-b=2
a^2-b^2=2(a+b)
2x+3-x+2=2(a+b)
(x+5)/2 = a+b
now add the original equation..
2= a-b
or 2a= 2+ x/2 + 5/2
4sqrt(2x+3)= x+9
square both sides..
16(2x+3)= x^2+18x + 81
x^2-14x+33=0
(x-3)(x-11)=0
square root sign 2x+3 - square root sign x-2 = 2. And the instructions are to solve the equation.
To solve the equation, you will need to follow these steps:
1. Start with the given equation:
√(2x+3) - √(x-2) = 2
2. Isolate one of the square roots. In this case, let's isolate √(2x+3):
√(2x+3) = 2 + √(x-2)
3. Square both sides to eliminate the square root:
(√(2x+3))^2 = (2 + √(x-2))^2
2x+3 = 4 + 4√(x-2) + (x-2)
2x+3 = x + 2 + 4√(x-2)
4. Simplify the equation by combining like terms:
2x - x + 3 - 2 = 4√(x-2)
x + 1 = 4√(x-2)
5. Square both sides again to eliminate the remaining square root:
(x + 1)^2 = (4√(x-2))^2
x^2 + 2x + 1 = 16(x-2)
x^2 + 2x + 1 = 16x - 32
6. Rearrange the equation to the standard quadratic form:
x^2 + 2x - 16x + 1 + 32 = 0
x^2 - 14x + 33 = 0
7. Factorize the quadratic equation by finding two numbers that multiply to give 33 and add up to -14. In this case, the numbers are -3 and -11, so we have:
(x - 3)(x - 11) = 0
8. Apply the zero-product property, which states that if a product of two factors equals zero, then at least one of the factors must be zero. So, we have two possibilities:
x - 3 = 0 => x = 3
x - 11 = 0 => x = 11
Therefore, the solutions to the equation √(2x+3) - √(x-2) = 2 are x = 3 and x = 11.