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The value of (-1 + √-3)2 + (-1 - √-3)2 is
- a)8
- b)4
- c)-4
- d)-2
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
The value of (-1 +√-3)2 + (-1 -√-3)2 isa)8b)4c)-4d)-2Corre...
The given expression can be simplified as follows:
-1 + √(-3) = -1 + i√3
-1 - √(-3) = -1 - i√3
Now, (-1 + i√3)2 = (-1)2 + 2*(-1)*(i√3) + (i√3)2
= 1 - 2i√3 - 3
= -2i√3 - 2
Similarly, (-1 - i√3)2 = (-1)2 + 2*(-1)*(-i√3) + (-i√3)2
= 1 + 2i√3 - 3
= 2i√3 - 2
Adding the two results together:
(-2i√3 - 2) + (2i√3 - 2) = -4
Therefore, the value of the expression is -4.
-1 + √(-3) = -1 + i√3
-1 - √(-3) = -1 - i√3
Now, (-1 + i√3)2 = (-1)2 + 2*(-1)*(i√3) + (i√3)2
= 1 - 2i√3 - 3
= -2i√3 - 2
Similarly, (-1 - i√3)2 = (-1)2 + 2*(-1)*(-i√3) + (-i√3)2
= 1 + 2i√3 - 3
= 2i√3 - 2
Adding the two results together:
(-2i√3 - 2) + (2i√3 - 2) = -4
Therefore, the value of the expression is -4.
Most Upvoted Answer
The value of (-1 +√-3)2 + (-1 -√-3)2 isa)8b)4c)-4d)-2Corre...
(-1 + √-3)² + (-1-√-3)²
Let a = - 1, b = √-3
(a + b)² + (a - b)² = a² + 2ab + b² + a² - 2ab + b²
=> 2a² + 2b² => 2 (a² + b²)
So, 2 [(-1)² + (√-3)²] => 2 * (1 + (-3)) => 2 * (1 - 3) => 2 * (-2) => - 4
So answer is - 4. Option c)
Let a = - 1, b = √-3
(a + b)² + (a - b)² = a² + 2ab + b² + a² - 2ab + b²
=> 2a² + 2b² => 2 (a² + b²)
So, 2 [(-1)² + (√-3)²] => 2 * (1 + (-3)) => 2 * (1 - 3) => 2 * (-2) => - 4
So answer is - 4. Option c)
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Community Answer
The value of (-1 +√-3)2 + (-1 -√-3)2 isa)8b)4c)-4d)-2Corre...
Understanding the Expression
To solve the expression (-1 + √-3)² + (-1 - √-3)², we first need to recognize that √-3 can be expressed as i√3, where i is the imaginary unit.
Substituting Values
1. Rewrite the terms:
- The first term becomes:
(-1 + i√3)²
- The second term becomes:
(-1 - i√3)²
Expanding Each Term
2. Expand both squares:
- For (-1 + i√3)²:
- (-1)² + 2(-1)(i√3) + (i√3)² = 1 - 2i√3 - 3 = -2 - 2i√3
- For (-1 - i√3)²:
- (-1)² + 2(-1)(-i√3) + (-i√3)² = 1 + 2i√3 - 3 = -2 + 2i√3
Combining the Results
3. Now, we add the two results:
- (-2 - 2i√3) + (-2 + 2i√3) = -2 - 2 + (-2i√3 + 2i√3) = -4 + 0i
Final Result
Thus, the combined expression simplifies to -4.
Conclusion
The value of (-1 + √-3)² + (-1 - √-3)² is -4, which corresponds to option 'C'.
To solve the expression (-1 + √-3)² + (-1 - √-3)², we first need to recognize that √-3 can be expressed as i√3, where i is the imaginary unit.
Substituting Values
1. Rewrite the terms:
- The first term becomes:
(-1 + i√3)²
- The second term becomes:
(-1 - i√3)²
Expanding Each Term
2. Expand both squares:
- For (-1 + i√3)²:
- (-1)² + 2(-1)(i√3) + (i√3)² = 1 - 2i√3 - 3 = -2 - 2i√3
- For (-1 - i√3)²:
- (-1)² + 2(-1)(-i√3) + (-i√3)² = 1 + 2i√3 - 3 = -2 + 2i√3
Combining the Results
3. Now, we add the two results:
- (-2 - 2i√3) + (-2 + 2i√3) = -2 - 2 + (-2i√3 + 2i√3) = -4 + 0i
Final Result
Thus, the combined expression simplifies to -4.
Conclusion
The value of (-1 + √-3)² + (-1 - √-3)² is -4, which corresponds to option 'C'.
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