(x-3i)-in-standard-form.jpeg "(x+3i)(x-3i) In Standard Form")
Simplifying (x+3i)(x-3i) into Standard Form
This expression represents the product of two complex conjugates. Let's break down the process of simplifying it into standard form, which is a + bi, where a and b are real numbers, and i is the imaginary unit (√-1).
Understanding Complex Conjugates
Complex conjugates are pairs of complex numbers that differ only in the sign of their imaginary parts. In this case, we have:
- (x + 3i) : This is the first complex number.
- (x - 3i) : This is the complex conjugate of the first number.
Simplifying the Expression
To simplify the product, we can use the difference of squares pattern: (a + b)(a - b) = a² - b²
Applying this pattern to our expression:
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Identify a and b:
- a = x
- b = 3i
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Substitute into the pattern: (x + 3i)(x - 3i) = x² - (3i)²
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Simplify: x² - (3i)² = x² - 9i²
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Remember that i² = -1: x² - 9i² = x² - 9(-1)
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Final simplification: x² - 9(-1) = x² + 9
Conclusion
Therefore, the simplified form of (x+3i)(x-3i) in standard form is x² + 9. This result demonstrates that the product of complex conjugates always results in a real number.