Write the product (6 − i)(6 + i) in the form a + bi.
Understand the Problem
The question asks to multiply two complex numbers (6 − i) and (6 + i) and express the result in the form a + bi, where a and b are real numbers. We will use the distributive property (FOIL) to find the product.
Answer
$37 + 0i$
Answer for screen readers
The final answer is $37 + 0i$.
Steps to Solve
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Identify the complex numbers We have two complex numbers: $$(6 - i)$$ and $$(6 + i)$$.
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Apply the distributive property (FOIL) Using the FOIL method, which stands for First, Outer, Inner, Last, we will multiply the two complex numbers:
- First: $6 \cdot 6 = 36$
- Outer: $6 \cdot i = 6i$
- Inner: $-i \cdot 6 = -6i$
- Last: $-i \cdot i = -i^2$, but since $i^2 = -1$, we have $-(-1) = 1$
- Combine the results Now, we combine all the results we found: $$ 36 + 6i - 6i + 1 $$
Notice that the $6i$ and $-6i$ cancel out.
- Simplify the expression The expression simplifies to: $$ 36 + 1 = 37 $$
Thus, the final result in the form $a + bi$ is: $$ 37 + 0i $$
The final answer is $37 + 0i$.
More Information
When multiplying complex numbers, if you apply the distributive property correctly, you may often notice that terms cancel out, particularly when dealing with conjugates like in this case: $(6 - i)$ and $(6 + i)$. The product of two conjugates always results in a real number.
Tips
- Forgetting to apply the rule that $i^2 = -1$. This can lead to getting the signs incorrect.
- Not combining like terms properly, especially when terms cancel out.
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