solving radical equations...
help!
5+¡în+1= n+4
To solve the given radical equation, we need to isolate the square root term and then solve for the variable.
Let's break it down step by step:
1. Start by subtracting 1 from both sides of the equation:
5 + √(n + 1) = n + 4 - 1
5 + √(n + 1) = n + 3
2. Next, subtract 5 from both sides of the equation to isolate the square root term:
5 - 5 + √(n + 1) = n + 3 - 5
√(n + 1) = n - 2
3. To eliminate the square root, square both sides of the equation:
(√(n + 1))^2 = (n - 2)^2
n + 1 = n^2 - 4n + 4
4. Simplify the right side of the equation by expanding and rearranging:
n^2 - 4n + 4 = n + 1
n^2 - 5n + 3 = 0
5. Move all terms to one side to form a quadratic equation:
n^2 - 5n + 3 = 0
6. To solve the quadratic equation, you can either factor, complete the square or use the quadratic formula. In this case, the equation cannot be easily factored, so we'll use the quadratic formula:
The quadratic formula is given by:
n = (-b ± √(b^2 - 4ac))/(2a)
Here, a = 1, b = -5, and c = 3.
Substitute these values into the quadratic formula:
n = (-(-5) ± √((-5)^2 - 4(1)(3)))/(2(1))
n = (5 ± √(25 - 12))/2
n = (5 ± √(13))/2
So, the solutions to the given radical equation are:
n = (5 + √13)/2 or n = (5 - √13)/2