Answer :
To solve the equation \(\sqrt{4i + 5} = 3 - \sqrt{t + 5}\), let’s go through the steps:
1. Start with the given equation:
[tex]\[ \sqrt{4i + 5} = 3 - \sqrt{t + 5} \][/tex]
2. Isolate the square roots on one side:
[tex]\[ \sqrt{4i + 5} + \sqrt{t + 5} = 3 \][/tex]
3. Square both sides to eliminate the square roots:
[tex]\[ (\sqrt{4i + 5} + \sqrt{t + 5})^2 = 3^2 \][/tex]
4. Expand the left-hand side using the binomial theorem:
[tex]\[ (4i + 5) + 2\sqrt{(4i + 5)(t + 5)} + (t + 5) = 9 \][/tex]
5. Combine like terms:
[tex]\[ 4i + t + 10 + 2\sqrt{(4i + 5)(t + 5)} = 9 \][/tex]
6. Move the constant term to the right-hand side:
[tex]\[ 4i + t + 2\sqrt{(4i + 5)(t + 5)} = -1 \][/tex]
Now we observe that the equation involves a square root term, a sum of terms, and a negative value on one side, which makes it critical to check possible solutions directly.
7. Test the candidate \(i = -1\):
- Calculate \(\sqrt{4(-1) + 5}\):
[tex]\[ \sqrt{4(-1) + 5} = \sqrt{-4 + 5} = \sqrt{1} = 1 \][/tex]
- Calculate \(3 - \sqrt{(-1) + 5}\):
[tex]\[ 3 - \sqrt{-1 + 5} = 3 - \sqrt{4} = 3 - 2 = 1 \][/tex]
- Both sides are equal:
[tex]\[ \sqrt{4(-1) + 5} = 1 \quad \text{and} \quad 3 - \sqrt{(-1) + 5} = 1 \][/tex]
Therefore, \(i = -1\) satisfies the equation.
8. Test the candidate \(i = 11\):
- Calculate \(\sqrt{4(11) + 5}\):
[tex]\[ \sqrt{4(11) + 5} = \sqrt{44 + 5} = \sqrt{49} = 7 \][/tex]
- Calculate \(3 - \sqrt{11 + 5}\):
[tex]\[ 3 - \sqrt{11 + 5} = 3 - \sqrt{16} = 3 - 4 = -1 \][/tex]
- Both sides are not equal:
[tex]\[ \sqrt{4(11) + 5} = 7 \quad \text{and} \quad 3 - \sqrt{11 + 5} = -1 \][/tex]
Thus, \(i = 11\) does not satisfy the equation.
After considering all candidates, the solution to the equation is:
[tex]\(\boxed{i = -1}\)[/tex]
1. Start with the given equation:
[tex]\[ \sqrt{4i + 5} = 3 - \sqrt{t + 5} \][/tex]
2. Isolate the square roots on one side:
[tex]\[ \sqrt{4i + 5} + \sqrt{t + 5} = 3 \][/tex]
3. Square both sides to eliminate the square roots:
[tex]\[ (\sqrt{4i + 5} + \sqrt{t + 5})^2 = 3^2 \][/tex]
4. Expand the left-hand side using the binomial theorem:
[tex]\[ (4i + 5) + 2\sqrt{(4i + 5)(t + 5)} + (t + 5) = 9 \][/tex]
5. Combine like terms:
[tex]\[ 4i + t + 10 + 2\sqrt{(4i + 5)(t + 5)} = 9 \][/tex]
6. Move the constant term to the right-hand side:
[tex]\[ 4i + t + 2\sqrt{(4i + 5)(t + 5)} = -1 \][/tex]
Now we observe that the equation involves a square root term, a sum of terms, and a negative value on one side, which makes it critical to check possible solutions directly.
7. Test the candidate \(i = -1\):
- Calculate \(\sqrt{4(-1) + 5}\):
[tex]\[ \sqrt{4(-1) + 5} = \sqrt{-4 + 5} = \sqrt{1} = 1 \][/tex]
- Calculate \(3 - \sqrt{(-1) + 5}\):
[tex]\[ 3 - \sqrt{-1 + 5} = 3 - \sqrt{4} = 3 - 2 = 1 \][/tex]
- Both sides are equal:
[tex]\[ \sqrt{4(-1) + 5} = 1 \quad \text{and} \quad 3 - \sqrt{(-1) + 5} = 1 \][/tex]
Therefore, \(i = -1\) satisfies the equation.
8. Test the candidate \(i = 11\):
- Calculate \(\sqrt{4(11) + 5}\):
[tex]\[ \sqrt{4(11) + 5} = \sqrt{44 + 5} = \sqrt{49} = 7 \][/tex]
- Calculate \(3 - \sqrt{11 + 5}\):
[tex]\[ 3 - \sqrt{11 + 5} = 3 - \sqrt{16} = 3 - 4 = -1 \][/tex]
- Both sides are not equal:
[tex]\[ \sqrt{4(11) + 5} = 7 \quad \text{and} \quad 3 - \sqrt{11 + 5} = -1 \][/tex]
Thus, \(i = 11\) does not satisfy the equation.
After considering all candidates, the solution to the equation is:
[tex]\(\boxed{i = -1}\)[/tex]
