%5E2-(2x-3i)%5E2.jpeg "(x+3i)^2-(2x-3i)^2")
Expanding and Simplifying (x + 3i)^2 - (2x - 3i)^2
This problem involves expanding and simplifying a complex expression. Let's break it down step-by-step:
1. Expanding the Squares
We start by expanding the squares using the formula (a + b)^2 = a^2 + 2ab + b^2 and (a - b)^2 = a^2 - 2ab + b^2.
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Expanding (x + 3i)^2: (x + 3i)^2 = x^2 + 2(x)(3i) + (3i)^2 = x^2 + 6xi + 9i^2
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Expanding (2x - 3i)^2: (2x - 3i)^2 = (2x)^2 - 2(2x)(3i) + (3i)^2 = 4x^2 - 12xi + 9i^2
2. Simplifying with i^2 = -1
Remember that the imaginary unit 'i' is defined as the square root of -1, so i^2 = -1. We can substitute this into our expanded expressions:
- (x + 3i)^2: x^2 + 6xi - 9
- (2x - 3i)^2: 4x^2 - 12xi - 9
3. Combining the Expanded Terms
Now we can subtract the simplified expressions:
(x^2 + 6xi - 9) - (4x^2 - 12xi - 9)
Simplify by distributing the negative sign:
x^2 + 6xi - 9 - 4x^2 + 12xi + 9
Combine like terms:
-3x^2 + 18xi
4. Final Result
Therefore, the simplified form of (x + 3i)^2 - (2x - 3i)^2 is -3x^2 + 18xi.
This expression is a complex number, with a real part of -3x^2 and an imaginary part of 18x.