Lesson 2: Does Every Complex Number Have a Square Root?

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 PRECALCULUS AND ADVANCED TOPICS

Student Outcomes . Students apply their understanding of polynomial identities that have been extended to the complex numbers to find the square roots of complex numbers.

Lesson Notes In Precalculus Module 1, students used the polar form of a complex number to find powers and roots of complex numbers. However, nearly all of the examples used in those lessons involved complex numbers with arguments that 휋 휋 휋 were multiples of special angles ( , , and ). In this lesson, we return to the rectangular form of a complex number to 4 3 2 show algebraically that we can find the square roots of any complex number without having to express it first in polar form. Students use the properties of complex numbers and the fundamental theorem of algebra to find the square roots of any complex number by creating and solving polynomial equations (N-NC.C.8 and N-NC.C.9). While solving these equations, students see that we arrive at polynomial identities with two factors that guarantee two roots to the equation. Throughout the lesson, students use algebraic properties to justify their reasoning (MP.3) and examine the structure of expressions to support their solutions and make generalizations (MP.7 and MP.8).

Classwork

Opening (5 minutes) Scaffolding: Organize the class into groups of 3–5 students. Start by displaying the . Create an anchor chart with the question shown below. Give students time to consider an answer to this following formulas: question on their own, and then allow them to discuss it in small groups. For any complex number 푧, Have a representative from each group briefly summarize their small group 푧푛 = 푟푛(cos(푛휃) + 푖 sin(푛휃)). discussions. The 푛th roots of 푧 = 푟푒푖휃 are given by . Does every complex number have a square root? If yes, provide at 휃 2휋푘 휃 2휋푘 least two examples. If no, explain why not. 푛 푟 (cos ( + ) + 푖 sin ( + )) √ 푛 푛 푛 푛  If we think about transformations, then squaring a complex for integers 푘 and 푛 such that 푛 > 0 number dilates a complex number by the modulus and and 0 ≤ 푘 < 푛. rotates it by the argument. Thus, taking the square root of a complex number should divide the argument by 2 and . Ask students to explain the formulas have a modulus equal to the square root of the original above for given values of 푛, 푟, and 휃. 휋 modulus. For example, substitute for 휃, 1 for 푟, 2 For example, the complex number 푖 would have a square and 2 for 푛 into the formula for the 푛th root with modulus equal to 1 and argument equal to 45°, roots of 푧, and ask students what the √2 √2 expressions represent (the square root which is + 푖. 2 2 of 푖).

Lesson 2: Does Every Complex Number Have a Square Root? 27

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 PRECALCULUS AND ADVANCED TOPICS

Student discussions and responses should reveal how much they recall from their work in Module 1 Lessons 18 and 19. In those lessons, students used the polar form of a complex number to find the 푛th roots of a complex number in polar form. Based on student responses to this question, the teacher may need to briefly review the polar form of a complex number and the formulas developed in Lessons 18 and 19 for finding powers of a complex number 푧 and roots of a complex number 푧.

Exercises 1–6 (10 minutes) Students should work these exercises with their group. As students work, be sure to circulate around the classroom to monitor the groups’ progress. Pause as needed for whole-group discussion and debriefing, especially if a majority of the groups are struggling to make progress.

Exercises 1–6

1. Use the geometric effect of complex multiplication to describe how to calculate a square root of 풛 = ퟏퟏퟗ + ퟏퟐퟎ풊.

The square root of this number would have a modulus equal to the square root of |ퟏퟏퟗ + ퟏퟐퟎ풊| and an argument 퐚퐫퐠 (ퟏퟏퟗ+ퟏퟐퟎ풊) equal to . ퟐ

2. Calculate an estimate of a square root of ퟏퟏퟗ + ퟏퟐퟎ풊.

|풛| = √ퟏퟏퟗퟐ + ퟏퟐퟎퟐ = ퟏퟔퟗ ퟏퟐퟎ 퐚퐫퐠(풛) = 퐚퐫퐜퐭퐚퐧 ( ) ퟏퟏퟗ

ퟏ ퟏퟐퟎ The square root’s modulus is √ퟏퟔퟗ or ퟏퟑ, and the square root’s argument is 퐚퐫퐜퐭퐚퐧 ( ). The square root is ퟐ ퟏퟏퟗ close to ퟏퟑ(퐜퐨퐬(ퟐퟐ. ퟔ°) + 풊 퐬퐢퐧(ퟐퟐ. ퟔ°)) or ퟏퟐ + ퟒ. ퟗퟗ풊.

If students get stuck on the next exercises, lead a short discussion to help set the stage for establishing that every complex number has two square roots that are opposites of each other. . What are the square roots of 4?  The square roots of 4 are 2 and −2 because (2)2 = 4 and (−2)2 = 4. . What are the square roots of 5? 2 5  The square roots of 5 are √5 and −√5 because (√5) = 5 and (−√5) = 5.

3. Every real number has two square roots. Explain why.

The fundamental theorem of algebra guarantees that a second-degree polynomial equation has two solutions. To find the square roots of a real number 풃, we need to solve the equation 풛ퟐ = 풃, which is a second-degree polynomial equation. Thus, the two solutions of 풛ퟐ = 풃 are the two square roots of 풃. If 풂 is one of the square roots, then – 풂 is MP.3 the other.

4. Provide a convincing argument that every complex number must also have two square roots.

By the same reasoning, if 풘 is a complex number, then the polynomial equation 풛ퟐ = 풘 has two solutions. The two solutions to this quadratic equation are the square roots of 풘. If 풂 + 풃풊 is one square root, then 풂 − 풃풊 is the other.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 PRECALCULUS AND ADVANCED TOPICS

5. Explain how the polynomial identity 풙ퟐ − 풃 = (풙 − √풃)(풙 + √풃) relates to the argument that every number has two square roots.

To solve 풙ퟐ = 풃, we can solve 풙ퟐ − 풃 = ퟎ. Since this quadratic equation has two distinct solutions, we can find two MP.3 square roots of 풃. The two square roots are opposites of each other.

6. What is the other square root of ퟏퟏퟗ + ퟏퟐퟎ풊?

It would be the opposite of ퟏퟐ + ퟓ풊, which is the complex number, −ퟏퟐ − ퟓ풊.

Example 1 (10 minutes): Find the Square Roots of ퟏퟏퟗ + ퟏퟐퟎ풊 The problem with using the polar form of a complex number to find its square roots is Scaffolding: that the argument of these numbers is not an easily recognizable number unless we . Students may benefit from pick our values of 푎 and 푏 very carefully, such as 1 + √3푖. practice and review of Recall from the last lesson that we proved using complex numbers that the equation factoring fourth-degree 푥2 = 1 had exactly two solutions. Here is another approach to finding both square polynomials. See Algebra II roots of a complex number that involves creating and solving a system of equations. Module 1. Some sample The solutions to these equations provide a way to define the square roots of a problems are provided below. complex number. Students have to solve a fourth-degree polynomial equation that 푥4 − 2푥2 − 8 has both real and imaginary solutions by factoring using polynomial identities. 푥4 − 9푥2 − 112 푥4 − 푥2 − 12 Example: Find the Square Roots of ퟏퟏퟗ + ퟏퟐퟎ풊 . Post the following identities on Let 풘 = 풑 + 풒풊 be the square root of ퟏퟏퟗ + ퟏퟐퟎ풊. Then the board: 2 2 풘ퟐ = ퟏퟏퟗ + ퟏퟐퟎ풊 푎 − 푏 = (푎 + 푏)(푎 − 푏) 푎2 + 푏2 = (푎 + 푏푖)(푎 − 푏푖). and . Students could use technology ퟐ (풑 + 풒풊) = ퟏퟏퟗ + ퟏퟐퟎ풊. like Desmos to quickly find real a. Expand the left side of this equation. number solutions to equations like those in part (b) by 풑ퟐ − 풒ퟐ + ퟐ풑풒풊 = ퟏퟏퟗ + ퟏퟐퟎ풊 graphing each side and finding the 푥-coordinates of the b. Equate the real and imaginary parts, and solve for 풑 and 풒. intersection points. 풑ퟐ − 풒ퟐ = ퟏퟏퟗ and ퟐ풑풒 = ퟏퟐퟎ. Solving for 풒 and substituting gives

ퟔퟎ ퟐ 풑ퟐ − ( ) = ퟏퟏퟗ. 풑

Multiplying by 풑ퟐ gives the equation

풑ퟒ − ퟏퟏퟗ풑ퟐ − ퟑퟔퟎퟎ = ퟎ. And this equation can be solved by factoring.

(풑ퟐ + ퟐퟓ)(풑ퟐ − ퟏퟒퟒ) = ퟎ We now have two polynomial expressions that we know how to factor: the sum and difference of squares.

(풑 + ퟓ풊)(풑 − ퟓ풊)(풑 − ퟏퟐ)(풑 + ퟏퟐ) = ퟎ The solutions are ퟓ풊, −ퟓ풊, ퟏퟐ, and −ퟏퟐ. Since 풑 must be a real number by the definition of complex number, we can choose ퟏퟐ or −ퟏퟐ for 풑. Using the equation ퟐ풑풒 = ퟏퟐퟎ, when 풑 = ퟏퟐ, 풒 = ퟓ, and when 풑 = −ퟏퟐ, 풒 = −ퟓ.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 PRECALCULUS AND ADVANCED TOPICS

c. What are the square roots of ퟏퟏퟗ + ퟏퟐퟎ풊?

Thus, the square roots of ퟏퟏퟗ + ퟏퟐퟎ풊 are ퟏퟐ + ퟓ풊 and −ퟏퟐ − ퟓ풊.

Debrief this example by discussing how this process could be generalized for any complex number 푧 = 푎 + 푏푖. Be sure to specifically mention that the polynomial identities (sum and difference of squares) each has two factors, so we get two solutions when setting those factors equal to zero. . If we solved this problem again using different values of 푎 and 푏 instead of 119 and 120, would we still get exactly two square roots of the form 푤 = 푝 + 푞푖 and 푤 = −푝 − 푞푖 that satisfy 푤2 = 푎 + 푏푖? Explain your reasoning.  Yes. When calculating 푤2, we would get a fourth-degree equation that would factor into two second- degree polynomial factors: One that is a difference of squares, and one that is a sum of squares. The difference of squares will have two linear factors with real coefficients giving us two values of 푝 that can be used to find the two square roots 푝 + 푞푖.

Exercises 7–9 (12 minutes) Students can work these exercises in groups. Be sure to have at least one group present their solution to Exercise 8. Before Exercise 7, use the discussion questions that follow if needed to activate students’ prior knowledge about complex conjugates.

Exercises 7–9

7. Use the method in the Example to find the square roots of ퟏ + √ퟑ풊.

풑ퟐ − 풒ퟐ = ퟏ and ퟐ풑풒 = √ퟑ Substituting and solving for 풑,

ퟐ √ퟑ 풑ퟐ − ( ) = ퟏ ퟐ풑 ퟒ풑ퟒ − ퟑ = ퟒ풑ퟐ ퟒ풑ퟒ − ퟒ풑ퟐ − ퟑ = ퟎ (ퟐ풑ퟐ − ퟑ)(ퟐ풑ퟐ + ퟏ) = ퟎ

ퟑ ퟑ 풑 = √ 풑 = −√ 풒 gives the real solutions ퟐ or ퟐ. The values of would then be

√ퟑ √ퟐ √ퟐ 풒 = = and 풒 = − . √ퟑ ퟐ ퟐ ퟐ ퟐ

√ퟔ √ퟐ √ퟔ √ퟐ The square roots of ퟏ + √ퟑ풊 are + 풊 and − − 풊. ퟐ ퟐ ퟐ ퟐ

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 PRECALCULUS AND ADVANCED TOPICS

8. Find the square roots of each complex number. a. ퟓ + ퟏퟐ풊

The square roots of ퟓ + ퟏퟐ풊 satisfy the equation (풑 + 풒풊)ퟐ = ퟓ + ퟏퟐ풊.

Expanding (풑 + 풒풊)ퟐ and equating the real and imaginary parts gives

풑ퟐ − 풒ퟐ = ퟓ ퟐ풑풒 = ퟏퟐ.

ퟔ 풒 = 풑ퟐ − 풒ퟐ = ퟓ Substituting 풑 into gives

ퟔ ퟐ 풑ퟐ − ( ) = ퟓ 풑 풑ퟒ − ퟑퟔ = ퟓ풑ퟐ 풑ퟒ − ퟓ풑ퟐ − ퟑퟔ = ퟎ (풑ퟐ − ퟗ)(풑ퟐ + ퟒ) = ퟎ.

The one positive real solution to this equation is ퟑ. Let 풑 = ퟑ. Then 풒 = ퟐ. A square root of ퟓ + ퟏퟐ풊 is ퟑ + ퟐ풊.

The other square root is when 풑 = −ퟑ and 풒 = −ퟐ. Therefore, −ퟑ − ퟐ풊 is the other square root of ퟓ + ퟏퟐ풊.

b. ퟓ − ퟏퟐ풊

The square roots of ퟓ − ퟏퟐ풊 satisfy the equation (풑 + 풒풊)ퟐ = ퟓ − ퟏퟐ풊.

Expanding (풑 + 풒풊)ퟐ and equating the real and imaginary parts gives

풑ퟐ − 풒ퟐ = ퟓ ퟐ풑풒 = −ퟏퟐ.

ퟔ 풒 = − 풑ퟐ − 풒ퟐ = ퟓ Substituting 풑 into gives

ퟔ ퟐ 풑ퟐ − (− ) = ퟓ 풑 풑ퟒ − ퟑퟔ = ퟓ풑ퟐ 풑ퟒ − ퟓ풑ퟐ − ퟑퟔ = ퟎ (풑ퟐ − ퟗ)(풑ퟐ + ퟒ) = ퟎ.

The one positive real solution to this equation is ퟑ. Let 풑 = ퟑ. Then 풒 = −ퟐ. A square root of ퟓ − ퟏퟐ풊 is ퟑ − ퟐ풊.

The other square root is when 풑 = −ퟑ and 풒 = ퟐ. Therefore, −ퟑ + ퟐ풊 is the other square root of ퟓ − ퟏퟐ풊.

. What do we call complex numbers of the form 푎 + 푏푖 and 푎 − 푏푖?  They are called complex conjugates. . Based on Exercise 6, how are square roots of conjugates related? Why do you think this relationship exists?  It appears that the square roots are opposites. When we solved the equation, the only difference was 6 that 2푝푞 = −12, which ended up not mattering when we squared − . 푝 . What is the conjugate of 119 + 120푖?  The conjugate is 119 − 120푖.

Lesson 2: Does Every Complex Number Have a Square Root? 31

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 PRECALCULUS AND ADVANCED TOPICS

. What do you think the square roots of 119 − 120푖 would be? Explain your reasoning.  The square roots of 119 − 120푖 should be the conjugates of the square roots of 119 + 120푖; the square roots should be 12 − 5푖 and −12 + 5푖.

9. Show that if 풑 + 풒풊 is a square root of 풛 = 풂 + 풃풊, then 풑 − 풒풊 is a square root of the conjugate of 풛, 풛̅ = 풂 − 풃풊. a. Explain why (풑 + 풒풊)ퟐ = 풂 + 풃풊.

If 풑 + 풒풊 is a square root of 풂 + 풃풊, then it must satisfy the definition of a square root. The square root of a number raised to the second power should equal the number.

b. What do 풂 and 풃 equal in terms of 풑 and 풒?

Expanding (풑 + 풒풊)ퟐ = 풑ퟐ − 풒ퟐ + ퟐ풑풒풊. Thus, 풂 = 풑ퟐ − 풒ퟐ and 풃 = ퟐ풑풒.

c. Calculate (풑 − 풒풊)ퟐ. What is the real part, and what is the imaginary part?

(풑 − 풒풊)ퟐ = 풑ퟐ − 풒ퟐ − ퟐ풑풒풊 The real part is 풑ퟐ − 풒ퟐ, and the imaginary part is −ퟐ풑풒.

d. Explain why (풑 − 풒풊)ퟐ = 풂 − 풃풊.

From part (c), 풑ퟐ − 풒ퟐ = 풂 and ퟐ풑풒 = 풃. Substituting,

(풑 − 풒풊)ퟐ = 풑ퟐ − 풒ퟐ − ퟐ풑풒풊 = 풂 − 풃풊.

Closing (3 minutes) Students can respond to the following questions either in writing or by discussing them with a partner. . How are square roots of complex numbers found by solving a polynomial equation?  If 푝 + 푞푖 is a square root of a complex number 푎 + 푏푖, then (푝 + 푞푖)2 = 푎 + 푏푖. Expanding the expression on the left and equating the real and imaginary parts leads to equations 푝2 − 푞2 = 푎 and 2푝푞 = 푏. Then, solve this resulting system of equations for 푝 and 푞. . Explain why the fundamental theorem of algebra guarantees that every complex number has two square roots.  According to the fundamental theorem of algebra, every second-degree polynomial equation factors into linear terms. The square roots of a complex number 푤 are solutions to the equation 푧2 = 푤, which can be rewritten as 푧2 − 푤 = 0. This equation has two solutions, each of which is a square root of 푤.

Lesson Summary

The square roots of a complex number 풂 + 풃풊 are of the form 풑 + 풒풊 and −풑 − 풒풊 and can be found by solving the equations 풑ퟐ − 풒ퟐ = 풂 and ퟐ풑풒 = 풃.

Exit Ticket (5 minutes)

Lesson 2: Does Every Complex Number Have a Square Root? 32

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 PRECALCULUS AND ADVANCED TOPICS

Name Date

Lesson 2: Does Every Complex Number Have a Square Root?

Exit Ticket

1. Find the two square roots of 5 − 12푖.

2. Find the two square roots of 3 − 4푖.

Lesson 2: Does Every Complex Number Have a Square Root? 33

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 PRECALCULUS AND ADVANCED TOPICS

Exit Ticket Sample Solutions

1. Find the two square roots of ퟓ − ퟏퟐ풊.

(풑 + 풒풊)ퟐ = ퟓ − ퟏퟐ풊, 풑ퟐ − 풒ퟐ + ퟐ풑풒풊 = ퟓ − ퟏퟐ풊, ퟔ 풑ퟐ − 풒ퟐ = ퟓ 풑풒 = −ퟔ 풒 = − , , 풑 ퟑퟔ 풑ퟐ − − ퟓ = ퟎ, 풑ퟒ − ퟓ풑ퟐ − ퟑퟔ = ퟎ, (풑ퟐ + ퟒ)(풑ퟐ − ퟗ) = ퟎ, 풑 = ±ퟑ, 풑ퟐ

풑 = ퟑ, 풒 = −ퟐ; 풑 = −ퟑ, 풒 = ퟐ

Therefore, the square roots are ퟑ − ퟐ풊 and −ퟑ + ퟐ풊.

2. Find the two square roots of ퟑ − ퟒ풊.

Let the square roots have the form 풑 + 풒풊. Then (풑 + 풒풊)ퟐ = ퟑ − ퟒ풊. This gives 풑ퟐ − 풒ퟐ = ퟑ and ퟐ풑풒 = −ퟒ. Then ퟐ ퟒ 풑 = − , so 풑ퟐ − 풒ퟐ = − 풒ퟐ = ퟑ, and 풒ퟒ + ퟑ풒ퟐ − ퟒ = ퟎ. This expression factors into 풒 풒ퟐ (풒 + ퟏ)(풒 − ퟏ)(풒 + ퟐ풊)(풒 − ퟐ풊) = ퟎ, and we see that the only real solutions are 풒 = ퟏ and 풒 = −ퟏ. If 풒 = ퟏ, then 풑 = −ퟐ, and if 풒 = −ퟏ, then 풑 = ퟐ. Thus, the two square roots of ퟑ − ퟒ풊 are −ퟐ + 풊 and ퟐ − 풊.

Problem Set Sample Solutions

Find the two square roots of each complex number by creating and solving polynomial equations.

1. 풛 = ퟏퟓ − ퟖ풊

(풑 + 풒풊)ퟐ = ퟏퟓ − ퟖ풊, 풑ퟐ − 풒ퟐ + ퟐ풑풒풊 = ퟏퟓ − ퟖ풊, ퟒ 풑ퟐ − 풒ퟐ = ퟏퟓ 풑풒 = −ퟒ 풒 = − , , 풑 ퟏퟔ 풑ퟐ − − ퟏퟓ = ퟎ, 풑ퟒ − ퟏퟓ풑ퟐ − ퟏퟔ = ퟎ, (풑ퟐ + ퟏ)(풑ퟐ − ퟏퟔ) = ퟎ, 풑 = ±ퟒ, 풑ퟐ

풑 = ퟒ, 풒 = −ퟏ; 풑 = −ퟒ, 풒 = ퟏ

Therefore, the square roots of ퟏퟓ − ퟖ풊 are ퟒ − 풊 and −ퟒ + 풊.

2. 풛 = ퟖ − ퟔ풊

(풑 + 풒풊)ퟐ = ퟖ − ퟔ풊, 풑ퟐ − 풒ퟐ + ퟐ풑풒풊 = ퟖ − ퟔ풊, ퟑ 풑ퟐ − 풒ퟐ = ퟖ 풑풒 = −ퟑ 풒 = − , , 풑 ퟗ 풑ퟐ − − ퟖ = ퟎ, 풑ퟒ − ퟖ풑ퟐ − ퟗ = ퟎ, (풑ퟐ + ퟏ)(풑ퟐ − ퟗ) = ퟎ, 풑 = ±ퟑ, 풑ퟐ

풑 = ퟑ, 풒 = −ퟏ; 풑 = −ퟑ, 풒 = ퟏ

Therefore, the square roots of ퟖ − ퟔ풊 are ퟑ − 풊 and −ퟑ + 풊.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 PRECALCULUS AND ADVANCED TOPICS

3. 풛 = −ퟑ + ퟒ풊

(풑 + 풒풊)ퟐ = −ퟑ + ퟒ풊, 풑ퟐ − 풒ퟐ + ퟐ풑풒풊 = −ퟑ + ퟒ풊, ퟐ 풑ퟐ − 풒ퟐ = −ퟑ 풑풒 = ퟐ 풒 = , , 풑 ퟒ 풑ퟐ − + ퟑ = ퟎ, 풑ퟒ + ퟑ풑ퟐ − ퟒ = ퟎ, (풑ퟐ + ퟒ)(풑ퟐ − ퟏ) = ퟎ, 풑 = ±ퟏ, 풑ퟐ 풑 = ퟏ, 풒 = ퟐ; 풑 = −ퟏ, 풒 = −ퟐ

Therefore, the square roots of −ퟑ + ퟒ풊 are ퟏ + ퟐ풊 and −ퟏ − ퟐ풊.

4. 풛 = −ퟓ − ퟏퟐ풊

(풑 + 풒풊)ퟐ = −ퟓ − ퟏퟐ풊, 풑ퟐ − 풒ퟐ + ퟐ풑풒풊 = −ퟓ − ퟏퟐ풊, −ퟔ 풑ퟐ − 풒ퟐ = −ퟓ 풑풒 = −ퟔ 풒 = , , 풑 ퟔ 풑ퟐ − + ퟓ = ퟎ, 풑ퟒ + ퟓ풑ퟐ − ퟑퟔ = ퟎ, (풑ퟐ + ퟗ)(풑ퟐ − ퟒ) = ퟎ, 풑 = ±ퟐ, 풑ퟐ

풑 = ퟐ, 풒 = −ퟑ; 풑 = −ퟐ, 풒 = ퟑ

Therefore, the square roots of −ퟓ − ퟏퟐ풊 are ퟐ − ퟑ풊 and −ퟐ + ퟑ풊.

5. 풛 = ퟐퟏ − ퟐퟎ풊

(풑 + 풒풊)ퟐ = ퟐퟏ − ퟐퟎ풊, 풑ퟐ − 풒ퟐ + ퟐ풑풒풊 = ퟐퟏ − ퟐퟎ풊, ퟏퟎ 풑ퟐ − 풒ퟐ = ퟐퟏ 풑풒 = −ퟏퟎ 풒 = − , , 풑 ퟏퟎퟎ 풑ퟐ − − ퟐퟏ = ퟎ, 풑ퟒ − ퟐퟏ풑ퟐ − ퟏퟎퟎ = ퟎ, (풑ퟐ + ퟒ)(풑ퟐ − ퟐퟓ) = ퟎ, 풑 = ±ퟓ, 풑ퟐ

풑 = ퟓ, 풒 = −ퟐ; 풑 = −ퟓ, 풒 = ퟐ

Therefore, the square roots of ퟐퟏ − ퟐퟎ풊 are ퟓ − ퟐ풊 and −ퟓ + ퟐ풊.

6. 풛 = ퟏퟔ − ퟑퟎ풊

(풑 + 풒풊)ퟐ = ퟏퟔ − ퟑퟎ풊, 풑ퟐ − 풒ퟐ + ퟐ풑풒풊 = ퟏퟔ − ퟑퟎ풊, ퟏퟓ 풑ퟐ − 풒ퟐ = ퟏퟔ 풑풒 = −ퟏퟓ 풒 = − , , 풑 ퟐퟐퟓ 풑ퟐ − − ퟏퟔ = ퟎ, 풑ퟒ − ퟏퟔ풑ퟐ − ퟐퟐퟓ = ퟎ, (풑ퟐ + ퟗ)(풑ퟐ − ퟐퟓ) = ퟎ, 풑 = ±ퟓ, 풑ퟐ

풑 = ퟓ, 풒 = −ퟑ; 풑 = −ퟓ, 풒 = ퟑ

Therefore, the square roots of ퟏퟔ − ퟑퟎ풊 are ퟓ − ퟑ풊 and −ퟓ + ퟑ풊.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 PRECALCULUS AND ADVANCED TOPICS

7. 풛 = 풊

(풑 + 풒풊)ퟐ = ퟎ + 풊, 풑ퟐ − 풒ퟐ + ퟐ풑풒풊 = ퟎ + 풊, ퟏ ퟏ 풑ퟐ − 풒ퟐ = ퟎ 풑풒 = 풒 = , ퟐ, ퟐ풑

ퟏ √ퟐ 풑ퟐ − = ퟎ, ퟒ풑ퟒ − ퟏ = ퟎ, (ퟐ풑ퟐ + ퟏ)(ퟐ풑ퟐ − ퟏ) = ퟎ, 풑 = ± ퟒ풑ퟐ ퟐ

√ퟐ √ퟐ √ퟐ √ퟐ 풑 = 풒 = 풑 = − 풒 = − ퟐ , ퟐ ; ퟐ , ퟐ

√ퟐ √ퟐ √ퟐ √ퟐ Therefore, the square roots of 풊 are + 풊 and − − 풊. ퟐ ퟐ ퟐ ퟐ

A Pythagorean triple is a set of three positive integers 풂, 풃, and 풄 such that 풂ퟐ + 풃ퟐ = 풄ퟐ. Thus, these integers can be the lengths of the sides of a right triangle.

8. Show algebraically that for positive integers 풑 and 풒, if

풂 = 풑ퟐ − 풒ퟐ 풃 = ퟐ풑풒 풄 = 풑ퟐ + 풒ퟐ

then 풂ퟐ + 풃ퟐ = 풄ퟐ.

풂ퟐ + 풃ퟐ = (풑ퟐ − 풒ퟐ)ퟐ + (ퟐ풑풒)ퟐ = 풑ퟒ − ퟐ풑ퟐ풒ퟐ + 풒ퟒ + ퟒ풑ퟐ풒ퟐ = 풑ퟒ + ퟐ풑ퟐ풒ퟐ + 풒ퟒ = 풑ퟒ + ퟐ풑ퟐ풒ퟐ + 풒ퟒ = 풄ퟐ

9. Select two integers 풑 and 풒, use the formulas in Problem 8 to find 풂, 풃, and 풄, and then show those numbers satisfy the equation 풂ퟐ + 풃ퟐ = 풄ퟐ.

Let 풑 = ퟑ and 풒 = ퟐ. Calculate the values of 풂, 풃, and 풄.

풂 = ퟑퟐ − ퟐퟐ = ퟓ 풃 = ퟐ(ퟑ)(ퟐ) = ퟏퟐ 풄 = ퟑퟐ + ퟐퟐ = ퟏퟑ 풂ퟐ + 풃ퟐ = ퟓퟐ + ퟏퟐퟐ = ퟐퟓ + ퟏퟒퟒ = ퟏퟔퟗ = ퟏퟑퟐ = 풄ퟐ

10. Use the formulas from Problem 8, and find values for 풑 and 풒 that give the following famous triples. a. (ퟑ, ퟒ, ퟓ)

풂 = 풑ퟐ − 풒ퟐ = ퟑ, 풃 = ퟐ풑풒 = ퟒ, 풄 = 풑ퟐ + 풒ퟐ = ퟓ

ퟐ풑ퟐ = ퟖ, 풑 = ퟐ, 풒 = ퟏ

(풑ퟐ − 풒ퟐ)ퟐ + (ퟐ풑풒)ퟐ = (풑ퟐ − 풒ퟐ)ퟐ

ퟐ (ퟐퟐ − ퟏퟐ)ퟐ + (ퟐ(ퟐ)(ퟏ)) = (ퟐퟐ + ퟏퟐ)ퟐ

(ퟒ − ퟏ)ퟐ + ퟒퟐ = (ퟒ + ퟏ)ퟐ, (ퟑ)ퟐ + (ퟒ)ퟐ = (ퟓ)ퟐ

Lesson 2: Does Every Complex Number Have a Square Root? 36

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 PRECALCULUS AND ADVANCED TOPICS

b. (ퟓ, ퟏퟐ, ퟏퟑ)

풂 = 풑ퟐ − 풒ퟐ = ퟓ, 풃 = ퟐ풑풒 = ퟏퟐ, 풄 = 풑ퟐ + 풒ퟐ = ퟏퟑ

ퟐ풑ퟐ = ퟏퟖ, 풑 = ퟑ, 풒 = ퟐ

(풑ퟐ − 풒ퟐ)ퟐ + (ퟐ풑풒)ퟐ = (풑ퟐ − 풒ퟐ)ퟐ

ퟐ (ퟑퟐ − ퟐퟐ)ퟐ + (ퟐ(ퟑ)(ퟐ)) = (ퟑퟐ + ퟐퟐ)ퟐ

(ퟗ − ퟒ)ퟐ + ퟏퟐퟐ = (ퟗ + ퟒ)ퟐ, (ퟓ)ퟐ + (ퟏퟐ)ퟐ = (ퟏퟑ)ퟐ

c. (ퟕ, ퟐퟒ, ퟐퟓ)

풂 = 풑ퟐ − 풒ퟐ = ퟕ, 풃 = ퟐ풑풒 = ퟐퟒ, 풄 = 풑ퟐ + 풒ퟐ = ퟐퟓ,

ퟐ풑ퟐ = ퟑퟐ, 풑 = ퟒ, 풒 = ퟑ

(풑ퟐ − 풒ퟐ)ퟐ + (ퟐ풑풒)ퟐ = (풑ퟐ − 풒ퟐ)ퟐ

ퟐ (ퟒퟐ − ퟑퟐ)ퟐ + (ퟐ(ퟒ)(ퟑ)) = (ퟒퟐ + ퟑퟐ)ퟐ

(ퟏퟔ − ퟗ)ퟐ + ퟐퟒퟐ = (ퟏퟔ + ퟗ)ퟐ, (ퟕ)ퟐ + (ퟐퟒ)ퟐ = (ퟐퟓ)ퟐ

d. (ퟗ, ퟒퟎ, ퟒퟏ)

풂 = 풑ퟐ − 풒ퟐ = ퟗ, 풃 = ퟐ풑풒 = ퟒퟎ, 풄 = 풑ퟐ + 풒ퟐ = ퟒퟏ,

ퟐ풑ퟐ = ퟓퟎ, 풑 = ퟓ, 풒 = ퟒ

(풑ퟐ − 풒ퟐ)ퟐ + (ퟐ풑풒)ퟐ = (풑ퟐ − 풒ퟐ)ퟐ

ퟐ (ퟓퟐ − ퟒퟐ)ퟐ + (ퟐ(ퟓ)(ퟒ)) = (ퟓퟐ + ퟒퟐ)ퟐ

(ퟐퟓ − ퟏퟔ)ퟐ + ퟒퟎퟐ = (ퟐퟓ + ퟏퟔ)ퟐ, (ퟗ)ퟐ + (ퟒퟎ)ퟐ = (ퟒퟏ)ퟐ

11. Is it possible to write the Pythagorean triple (ퟔ, ퟖ, ퟏퟎ) in the form 풂 = 풑ퟐ − 풒ퟐ, 풃 = ퟐ풑풒, 풄 = 풑ퟐ + 풒ퟐ for some integers 풑 and 풒? Verify your answer.

풂 = 풑ퟐ − 풒ퟐ = ퟔ, 풃 = ퟐ풑풒 = ퟖ, 풄 = 풑ퟐ + 풒ퟐ = ퟏퟎ,

ퟐ풑ퟐ = ퟏퟔ, 풑ퟐ = ퟖ, 풑 = ±ퟐ√ퟐ, 풒 = ±√ퟐ

The Pythagorean triple (ퟔ, ퟖ, ퟏퟎ) cannot be written in the form 풂 = 풑ퟐ − 풒ퟐ, 풃 = ퟐ풑풒, 풄 = 풑ퟐ + 풒ퟐ, for any integers 풑 and 풒.

12. Choose your favorite Pythagorean triple (풂, 풃, 풄) that has 풂 and 풃 sharing only ퟏ as a common factor, for example (ퟑ, ퟒ, ퟓ), (ퟓ, ퟏퟐ, ퟏퟑ), or (ퟕ, ퟐퟒ, ퟐퟓ), … Find the square of the length of a square root of 풂 + 풃풊; that is, find |풑 + 풒풊|ퟐ, where 풑 + 풒풊 is a square root of 풂 + 풃풊. What do you observe?

For (ퟑ, ퟒ, ퟓ), 풂 = ퟑ, 풃 = ퟒ, 풂 + 풃풊 = (풑 + 풒풊)ퟐ = (−풑 − 풒풊)ퟐ, ퟐ 풂 + 풃풊 = ퟑ + ퟒ풊 = (풑 + 풒풊)ퟐ = 풑ퟐ − 풒ퟐ + ퟐ풑풒풊 풑ퟐ − 풒ퟐ = ퟑ ퟐ풑풒 = ퟒ 풒 = ; therefore, , , 풑 ퟒ 풑ퟐ − = ퟑ, 풑ퟒ − ퟑ풑ퟐ − ퟒ = ퟎ, (풑ퟐ + ퟏ)(풑ퟐ − ퟒ) = ퟎ, 풑 = ±ퟐ 풑ퟐ

풑 = ퟐ, 풒 = ퟏ; therefore, 풑 + 풒풊 = ퟐ + 풊.

풑 = −ퟐ, 풒 = −ퟏ; therefore, 풑 + 풒풊 = −ퟐ − 풊.

The two square roots of 풂 + 풃풊 are ퟐ + 풊 and – ퟐ − 풊. Both of these have length √ퟓ, so the squared length is ퟓ. This is the third value 풄 in the Pythagorean triple (풂, 풃, 풄).

Lesson 2: Does Every Complex Number Have a Square Root? 37