Answer:
To solve for x, we can start by squaring both sides of the equation:
(√x+3)² = (2√x-2)²
On the left side, we have:
(√x+3)² = x + 6√x + 9
On the right side, we have:
(2√x-2)² = (2√x)² - 2(2√x)(2) + 2² = 4x - 8√x + 4
So now we have:
x + 6√x + 9 = 4x - 8√x + 4
Grouping like terms, we can simplify this to:
10√x - 3x + 5 = 0
Now we can solve for x by isolating the radical term and squaring both sides:
10√x - 3x + 5 = 0
10√x = 3x - 5
(10√x)² = (3x - 5)²
100x = 9x² - 30x + 25
Moving all the terms to one side, we get:
9x² - 130x + 25 = 0
We can now solve this quadratic equation by factoring or using the quadratic formula. Factoring, we have:
(3x - 5)(3x - 5) = 0
So the solution is:
3x - 5 = 0
3x = 5
x = 5/3
Therefore, the solution to the equation √x+3 = 2√x-2 is x = 5/3.
Explanation: