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Sagot :
To solve the given equation [tex]\((a_i + b_i i)(a_i + b_i i) = 34 + 8i\)[/tex], we need to find the integers [tex]\(a_i\)[/tex] and [tex]\(b_i\)[/tex] that satisfy this equation.
### Step 1: Understand the Equation
The equation [tex]\((a_i + b_i i)^2 = 34 + 8i\)[/tex] can be expanded to:
[tex]\[ (a_i^2 - b_i^2) + 2a_i b_i i = 34 + 8i \][/tex]
This means we have to equate the real part and the imaginary part separately:
[tex]\[ a_i^2 - b_i^2 = 34 \quad \text{(1)} \][/tex]
[tex]\[ 2a_i b_i = 8 \quad \text{(2)} \][/tex]
### Step 2: Solve the Imaginary Part Equation
From equation (2), solve for [tex]\(a_i\)[/tex] and [tex]\(b_i\)[/tex]:
[tex]\[ 2a_i b_i = 8 \][/tex]
[tex]\[ a_i b_i = 4 \quad \text{(3)} \][/tex]
### Step 3: Substitute into the Real Part Equation
We will try different integer values of [tex]\(b_i\)[/tex] that fit equation (3):
1. [tex]\(b_i = 1\)[/tex]:
[tex]\[ a_i \cdot 1 = 4 \implies a_i = 4 \][/tex]
Substitute into equation (1):
[tex]\[ 4^2 - 1^2 = 16 - 1 = 15 \neq 34 \][/tex]
This combination does not work.
2. [tex]\(b_i = 2\)[/tex]:
[tex]\[ a_i \cdot 2 = 4 \implies a_i = 2 \][/tex]
Substitute into equation (1):
[tex]\[ 2^2 - 2^2 = 4 - 4 = 0 \neq 34 \][/tex]
This combination does not work.
3. [tex]\(b_i = 4\)[/tex]:
[tex]\[ a_i \cdot 4 = 4 \implies a_i = 1 \][/tex]
Substitute into equation (1):
[tex]\[ 1^2 - 4^2 = 1 - 16 = -15 \neq 34 \][/tex]
This combination does not work.
4. [tex]\(b_i = -1\)[/tex]:
[tex]\[ a_i \cdot (-1) = 4 \implies a_i = -4 \][/tex]
Substitute into equation (1):
[tex]\[ (-4)^2 - (-1)^2 = 16 - 1 = 15 \neq 34 \][/tex]
This combination does not work.
5. [tex]\(b_i = -2\)[/tex]:
[tex]\[ a_i \cdot (-2) = 4 \implies a_i = -2 \][/tex]
Substitute into equation (1):
[tex]\[ (-2)^2 - (-2)^2 = 4 - 4 = 0 \neq 34 \][/tex]
This combination does not work.
6. [tex]\(b_i = -4\)[/tex]:
[tex]\[ a_i \cdot (-4) = 4 \implies a_i = -1 \][/tex]
Substitute into equation (1):
[tex]\[ (-1)^2 - (-4)^2 = 1 - 16 = -15 \neq 34 \][/tex]
This combination does not work.
### Step 4: Consider Complex Solutions
Since integer values do not satisfy the equations, let's express the solutions in terms of [tex]\(b_i\)[/tex]:
Given equation (3):
[tex]\[ a_i = \frac{4}{b_i} \][/tex]
### Step 5: Analytical Solution
Insert into equation (1):
[tex]\[ \left(\frac{4}{b_i}\right)^2 - b_i^2 = 34 \][/tex]
[tex]\[ \frac{16}{b_i^2} - b_i^2 = 34 \][/tex]
[tex]\[ 16 - b_i^4 = 34b_i^2 \][/tex]
[tex]\[ b_i^4 + 34b_i^2 - 16 = 0 \][/tex]
This is a quadratic in [tex]\(b_i^2\)[/tex]. Let [tex]\(u = b_i^2\)[/tex]:
[tex]\[ u^2 + 34u - 16 = 0 \][/tex]
Solve this quadratic using the quadratic formula:
[tex]\[ u = \frac{-34 \pm \sqrt{34^2 + 64}}{2} \][/tex]
[tex]\[ u = \frac{-34 \pm \sqrt{1156 + 64}}{2} \][/tex]
[tex]\[ u = \frac{-34 \pm \sqrt{1220}}{2} \][/tex]
[tex]\[ u = \frac{-34 \pm \sqrt{4 \cdot 305}}{2} \][/tex]
[tex]\[ u = \frac{-34 \pm 2\sqrt{305}}{2} \][/tex]
[tex]\[ u = -17 \pm \sqrt{305} \][/tex]
Therefore:
[tex]\[ b_i^2 = -17 + \sqrt{305} \quad \text{or} \quad b_i^2 = -17 - \sqrt{305} \][/tex]
Since [tex]\(b_i^2\)[/tex] must be positive, we discard the negative square root. Hence, [tex]\(b_i^2 = -17 + \sqrt{305}\)[/tex].
### Step 6: Determine [tex]\(a_i\)[/tex] and [tex]\(b_i\)[/tex]
Based on previous steps, the complex solutions were found to be:
[tex]\[ (a_i, b_i) = (-i \cdot b_i \pm \sqrt{34 + 8i}, b_i) \][/tex]
The final possible solutions [tex]\(a_i\)[/tex] and [tex]\(b_i\)[/tex] can be expressed as:
[tex]\[ \boxed{((-i \cdot b_i - \sqrt{34 + 8i}, b_i), (-i \cdot b_i + \sqrt{34 + 8i}, b_i))} \][/tex]
### Step 1: Understand the Equation
The equation [tex]\((a_i + b_i i)^2 = 34 + 8i\)[/tex] can be expanded to:
[tex]\[ (a_i^2 - b_i^2) + 2a_i b_i i = 34 + 8i \][/tex]
This means we have to equate the real part and the imaginary part separately:
[tex]\[ a_i^2 - b_i^2 = 34 \quad \text{(1)} \][/tex]
[tex]\[ 2a_i b_i = 8 \quad \text{(2)} \][/tex]
### Step 2: Solve the Imaginary Part Equation
From equation (2), solve for [tex]\(a_i\)[/tex] and [tex]\(b_i\)[/tex]:
[tex]\[ 2a_i b_i = 8 \][/tex]
[tex]\[ a_i b_i = 4 \quad \text{(3)} \][/tex]
### Step 3: Substitute into the Real Part Equation
We will try different integer values of [tex]\(b_i\)[/tex] that fit equation (3):
1. [tex]\(b_i = 1\)[/tex]:
[tex]\[ a_i \cdot 1 = 4 \implies a_i = 4 \][/tex]
Substitute into equation (1):
[tex]\[ 4^2 - 1^2 = 16 - 1 = 15 \neq 34 \][/tex]
This combination does not work.
2. [tex]\(b_i = 2\)[/tex]:
[tex]\[ a_i \cdot 2 = 4 \implies a_i = 2 \][/tex]
Substitute into equation (1):
[tex]\[ 2^2 - 2^2 = 4 - 4 = 0 \neq 34 \][/tex]
This combination does not work.
3. [tex]\(b_i = 4\)[/tex]:
[tex]\[ a_i \cdot 4 = 4 \implies a_i = 1 \][/tex]
Substitute into equation (1):
[tex]\[ 1^2 - 4^2 = 1 - 16 = -15 \neq 34 \][/tex]
This combination does not work.
4. [tex]\(b_i = -1\)[/tex]:
[tex]\[ a_i \cdot (-1) = 4 \implies a_i = -4 \][/tex]
Substitute into equation (1):
[tex]\[ (-4)^2 - (-1)^2 = 16 - 1 = 15 \neq 34 \][/tex]
This combination does not work.
5. [tex]\(b_i = -2\)[/tex]:
[tex]\[ a_i \cdot (-2) = 4 \implies a_i = -2 \][/tex]
Substitute into equation (1):
[tex]\[ (-2)^2 - (-2)^2 = 4 - 4 = 0 \neq 34 \][/tex]
This combination does not work.
6. [tex]\(b_i = -4\)[/tex]:
[tex]\[ a_i \cdot (-4) = 4 \implies a_i = -1 \][/tex]
Substitute into equation (1):
[tex]\[ (-1)^2 - (-4)^2 = 1 - 16 = -15 \neq 34 \][/tex]
This combination does not work.
### Step 4: Consider Complex Solutions
Since integer values do not satisfy the equations, let's express the solutions in terms of [tex]\(b_i\)[/tex]:
Given equation (3):
[tex]\[ a_i = \frac{4}{b_i} \][/tex]
### Step 5: Analytical Solution
Insert into equation (1):
[tex]\[ \left(\frac{4}{b_i}\right)^2 - b_i^2 = 34 \][/tex]
[tex]\[ \frac{16}{b_i^2} - b_i^2 = 34 \][/tex]
[tex]\[ 16 - b_i^4 = 34b_i^2 \][/tex]
[tex]\[ b_i^4 + 34b_i^2 - 16 = 0 \][/tex]
This is a quadratic in [tex]\(b_i^2\)[/tex]. Let [tex]\(u = b_i^2\)[/tex]:
[tex]\[ u^2 + 34u - 16 = 0 \][/tex]
Solve this quadratic using the quadratic formula:
[tex]\[ u = \frac{-34 \pm \sqrt{34^2 + 64}}{2} \][/tex]
[tex]\[ u = \frac{-34 \pm \sqrt{1156 + 64}}{2} \][/tex]
[tex]\[ u = \frac{-34 \pm \sqrt{1220}}{2} \][/tex]
[tex]\[ u = \frac{-34 \pm \sqrt{4 \cdot 305}}{2} \][/tex]
[tex]\[ u = \frac{-34 \pm 2\sqrt{305}}{2} \][/tex]
[tex]\[ u = -17 \pm \sqrt{305} \][/tex]
Therefore:
[tex]\[ b_i^2 = -17 + \sqrt{305} \quad \text{or} \quad b_i^2 = -17 - \sqrt{305} \][/tex]
Since [tex]\(b_i^2\)[/tex] must be positive, we discard the negative square root. Hence, [tex]\(b_i^2 = -17 + \sqrt{305}\)[/tex].
### Step 6: Determine [tex]\(a_i\)[/tex] and [tex]\(b_i\)[/tex]
Based on previous steps, the complex solutions were found to be:
[tex]\[ (a_i, b_i) = (-i \cdot b_i \pm \sqrt{34 + 8i}, b_i) \][/tex]
The final possible solutions [tex]\(a_i\)[/tex] and [tex]\(b_i\)[/tex] can be expressed as:
[tex]\[ \boxed{((-i \cdot b_i - \sqrt{34 + 8i}, b_i), (-i \cdot b_i + \sqrt{34 + 8i}, b_i))} \][/tex]
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