1. √(x+4)=√(x-1)+1
2. 2√n+3=n
√(x+4)=√(x-1)+1
x+4 = (x-1) + 2√(x-1) + 1
4 = 2√(x-1)
16 = 4(x-1)
x = 5
check: √9 = √4+1
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2√n + 3 = n
2√n = n-3
4n = n^2 - 6n + 9
n^2 - 10n + 9 = 0
(n-1)(n-9) = 0
n = 1 or 9
check:
2√1 + 3 = 1? nope - spurious solution
2√9 + 3 = 9? ok
2- n=t^ ---> √n=t 2t+3=t^ t^-2t-3
D=4-(4*(-3))=16 t1=(-(-2)+√D)\2
t1=2+4\2 3 t2=(-(-2)-√D)\2 t2=-1
n=3 n=1 is not correct that √-1 is not true
1. To solve the equation √(x+4) = √(x-1) + 1, you can follow these steps:
Step 1: Square both sides of the equation to eliminate the square roots:
(√(x+4))^2 = (√(x-1) + 1)^2
Simplifying the left side:
x + 4 = (x-1) + 2√(x-1) + 1
Simplifying the right side:
x + 4 = x - 1 + 2√(x-1) + 1
x + 4 = x + 2√(x-1)
Step 2: Isolate the square root term:
Subtract x from both sides:
4 = 2√(x-1)
Step 3: Divide by 2 to isolate the square root term:
2 = √(x-1)
Step 4: Square both sides again to eliminate the square root:
(2)^2 = (√(x-1))^2
Simplifying:
4 = x - 1
Step 5: Isolate x by adding 1 to both sides:
4 + 1 = x
x = 5
Therefore, the solution to the equation is x = 5.
2. To solve the equation 2√n + 3 = n, you can follow these steps:
Step 1: Square both sides of the equation to eliminate the square root:
(2√n + 3)^2 = n^2
Simplifying the left side:
4n + 12√n + 9 = n^2
Step 2: Move all terms to one side to set the equation equal to zero:
n^2 - 4n - 12√n - 9 = 0
Step 3: Simplify the equation as much as possible. Since we cannot combine the terms involving n and √n, we can't solve it algebraically. However, we can use numerical methods, such as graphing or solving it using a calculator or software.
Using a graphing calculator or software, plot the equation y = n^2 - 4n - 12√n - 9. Find where it intersects the x-axis to find the solutions.
Therefore, to find the solutions to the equation 2√n + 3 = n, you will need to use numerical methods such as graphing or solving it using a calculator or software.