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Common Core: High School - Number and Quantity : Complex Conjugate: CCSS.Math.Content.HSN-CN.A.3

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Example Question #1 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Find the product of (2-i) $\displaystyle (2-i)$ and its complex conjugate.

Possible Answers:

2-4i $\displaystyle 2-4i$

2+4i $\displaystyle 2+4i$

$\displaystyle 4$

$\displaystyle 5$

$\displaystyle 2$

Correct answer:

$\displaystyle 5$

Explanation:

This question tests one's understanding on complex numbers and their corresponding conjugate. It is important to remember that when a complex number is multiplied by its conjugate, the imaginary numbers disappear. A complex number and its conjugate takes the form as follows:

$\\(a+bi)(a-bi)=a^2+abi-abi-b^2i^2=a^2-b^2(-1)=a^2+b^2$ $\displaystyle \\(a+bi)(a-bi)=a^2+abi-abi-b^2i^2=a^2-b^2(-1)=a^2+b^2$

Note: i^2=-1 $\displaystyle i^2=-1$

For the purpose of Common Core Standards, "find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers", falls within the Cluster A of "perform arithmetic operations with complex numbers" (CCSS.MATH.CONTENT.HSF.CN.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Using the formula above find the conjugate.

The formula states to keep the real portion and the imaginary portion but switch the sign between them.

Complex number: (2-i) $\displaystyle (2-i)$

Real portion: $\displaystyle 2$

Imaginary portion: $\displaystyle i$

Conjugate: (2+i) $\displaystyle (2+i)$

Step 2: Set up the multiplication problem between the complex number and its conjugate.

$\displaystyle (2-i)(2+i)$

Step 3: Multiply the two binomials together.

Distribute the first term in the first binomial to both terms in the second binomial. Then distribute the second term in the first binomial to each term in the second binomial.

$\\ 2(2)+2(i)=4+2i \\-i(2)+-i(i)=-2i-i^2$ $\displaystyle \\ 2(2)+2(i)=4+2i \\-i(2)+-i(i)=-2i-i^2$

From here combine the two new expressions together.

$\displaystyle 4+2i-2i-i^2$

Since i^2=-1 $\displaystyle i^2=-1$ the product becomes,

$\\4+2i-2i-(-1) \\4+1 \\5$ $\displaystyle \\4+2i-2i-(-1) \\4+1 \\5$

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Example Question #1 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Find the conjugate of the complex number.

6+2i $\displaystyle 6+2i$

Possible Answers:

6+2i $\displaystyle 6+2i$

6-2i $\displaystyle 6-2i$

$\displaystyle 4i$

$\displaystyle 8i$

Correct answer:

6-2i $\displaystyle 6-2i$

Explanation:

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Example Question #2 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Find the conjugate of the complex number.

32-4i $\displaystyle 32-4i$

Possible Answers:

28i $\displaystyle 28i$

$\displaystyle 4i$

32+4i $\displaystyle 32+4i$

32-4i $\displaystyle 32-4i$

Correct answer:

32+4i $\displaystyle 32+4i$

Explanation:

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Example Question #3 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Find the conjugate of the complex number.

$\displaystyle 7i$

Possible Answers:

-7i $\displaystyle -7i$

$\displaystyle i$

$\displaystyle 7i$

$\displaystyle 7$

Correct answer:

-7i $\displaystyle -7i$

Explanation:

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Example Question #4 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Find the conjugate of the complex number.

$\displaystyle 6$

Possible Answers:

$\displaystyle -6$

$\displaystyle 6i$

-6i $\displaystyle -6i$

$\displaystyle 6$

Correct answer:

$\displaystyle 6$

Explanation:

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Example Question #3 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Find the product of the complex number and its conjugate.

4+2i $\displaystyle 4+2i$

Possible Answers:

20-16i $\displaystyle 20-16i$

$12i^{2}$ $\displaystyle 12i^{2}$

$16-4i^{2}$ $\displaystyle 16-4i^{2}$

$\displaystyle 20$

Correct answer:

$\displaystyle 20$

Explanation:

$\left ( 4+2i \right )\left ( 4-2i \right )$ $\displaystyle \left ( 4+2i \right )\left ( 4-2i \right )$

$16-8i+8i-4i^{2}$ $\displaystyle 16-8i+8i-4i^{2}$

16+4 $\displaystyle 16+4$

$\displaystyle 20$

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Example Question #4 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Find the product of the complex number and its conjugate.

12-5i $\displaystyle 12-5i$

Possible Answers:

169 $\displaystyle 169$

144-25i $\displaystyle 144-25i$

$49i^{2}$ $\displaystyle 49i^{2}$

119 $\displaystyle 119$

Correct answer:

169 $\displaystyle 169$

Explanation:

$\left ( 12-5i \right )\left ( 12+5i \right )$ $\displaystyle \left ( 12-5i \right )\left ( 12+5i \right )$

$144+60i-60i-25i^{2}$ $\displaystyle 144+60i-60i-25i^{2}$

144+25 $\displaystyle 144+25$

169 $\displaystyle 169$

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Example Question #5 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Find the product of the complex number and its conjugate.

1-i $\displaystyle 1-i$

Possible Answers:

$1-i^{2}$ $\displaystyle 1-i^{2}$

$\displaystyle 0$

1-i $\displaystyle 1-i$

$\displaystyle 2$

Correct answer:

$\displaystyle 2$

Explanation:

$\left ( 1-i \right )\left ( 1+i \right )$ $\displaystyle \left ( 1-i \right )\left ( 1+i \right )$

$1+i-i-i^{2}$ $\displaystyle 1+i-i-i^{2}$

1+1 $\displaystyle 1+1$

$\displaystyle 2$

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Example Question #6 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Find the product of the complex number and its conjugate.

20i $\displaystyle 20i$

Possible Answers:

-20i $\displaystyle -20i$

$\displaystyle 20$

$\displaystyle -400$

400 $\displaystyle 400$

Correct answer:

400 $\displaystyle 400$

Explanation:

$20i\cdot -20i$ $\displaystyle 20i\cdot -20i$

$-400i^{2}$ $\displaystyle -400i^{2}$

$-400\cdot -1$ $\displaystyle -400\cdot -1$

400 $\displaystyle 400$

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Example Question #7 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Simplify.

$\frac{2+i}{4-i}$ $\displaystyle \frac{2+i}{4-i}$

Possible Answers:

$\frac{7+6i}{15}$ $\displaystyle \frac{7+6i}{15}$

$\frac{7+6i}{4-i}$ $\displaystyle \frac{7+6i}{4-i}$

$\displaystyle 2i$

$\frac{1}{2-1}$ $\displaystyle \frac{1}{2-1}$

Correct answer:

$\frac{7+6i}{15}$ $\displaystyle \frac{7+6i}{15}$

Explanation:

$\frac{2+i}{4-i}\cdot \frac{4+i}{4+i}$ $\displaystyle \frac{2+i}{4-i}\cdot \frac{4+i}{4+i}$

$=\frac{\left ( 2+i \right )\left ( 4+i \right )}{\left ( 4-i \right )\left ( 4+i \right )}$ $\displaystyle =\frac{\left ( 2+i \right )\left ( 4+i \right )}{\left ( 4-i \right )\left ( 4+i \right )}$

$=\frac{8+2i+4i+i^{2}}{16+4i-4i-i^{2}}$ $\displaystyle =\frac{8+2i+4i+i^{2}}{16+4i-4i-i^{2}}$

$=\frac{8+6i-1}{16-1}$ $\displaystyle =\frac{8+6i-1}{16-1}$

$=\frac{7+6i}{15}$ $\displaystyle =\frac{7+6i}{15}$

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Common Core: High School - Number and Quantity : Complex Conjugate: CCSS.Math.Content.HSN-CN.A.3

Study concepts, example questions & explanations for Common Core: High School - Number and Quantity

All Common Core: High School - Number and Quantity Resources

Example Questions

Example Question #1 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Example Question #1 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Example Question #2 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Example Question #3 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Example Question #4 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Example Question #3 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Example Question #4 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Example Question #5 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Example Question #6 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

Example Question #7 : Complex Conjugate: Ccss.Math.Content.Hsn Cn.A.3

All Common Core: High School - Number and Quantity Resources

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