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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #31 : Complex Imaginary Numbers

Simplify: $\frac{i}{i^2+2i-1}$

Possible Answers:

$-\frac{1}{4}i+\frac{1}{4}$

$-\frac{1}{8}i+\frac{1}{8}$

$-\frac{3}{8}i+\frac{1}{2}$

$-\frac{3}{4}i+\frac{3}{8}$

$-\frac{1}{2}i+\frac{1}{2}$

Correct answer:

$-\frac{1}{4}i+\frac{1}{4}$

Explanation:

Simplify the denominator. Recall that $i=\sqrt{-1}$ , and $i^2= i\cdot i = -1$ .

$\frac{i}{i^2+2i-1} =\frac{i}{(-1)+2i-1} =\frac{i}{2i-2}$

Multiply the top and bottom by the conjugate of the denominator. Use the FOIL method to simplify the denominator.

$\frac{i}{2i-2}\cdot \frac{2i+2}{2i+2} = \frac{2i^2+2i}{4i^2-4}$

Replace the values of the known imaginary terms.

$\frac{2i^2+2i}{4i^2-4} =\frac{2(-1)+2i}{4(-1)-4}=\frac{2i-2}{-8}$

Simplify this fraction.

The answer is: $-\frac{1}{4}i+\frac{1}{4}$

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Example Question #36 : Imaginary Numbers

Simplify: $\frac{10}{10+i}$

Possible Answers:

$-\frac{10}{101}i+\frac{100}{101}$

$-\frac{10}{101}i-\frac{100}{101}$

$-\frac{10}{101}i-\frac{101}{100}$

$\frac{1}{10}i-\frac{11}{10}$

$\frac{1}{10}i-\frac{10}{11}$

Correct answer:

$-\frac{10}{101}i+\frac{100}{101}$

Explanation:

In order to simplify this, we will need to multiply both the top and bottom by the conjugate of the denominator.

$\frac{10}{10+i} = \frac{10}{i+10}$

$\frac{10}{i+10} \times \frac{i-10}{i-10}$

Simplify the top and bottom. The bottom can be simplified by the FOIL method.

$\frac{10}{i+10} \times \frac{i-10}{i-10} = \frac{10i-100}{i^2-100}$

The value of $i^2= i\cdot i = -1$ . Replace the term.

$\frac{10i-100}{-1-100} = \frac{10i-100}{-101}$

Reduce this fraction.

The answer is: $-\frac{10}{101}i+\frac{100}{101}$

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Example Question #31 : Complex Imaginary Numbers

Simplify: $\frac{3i}{i+9}$

Possible Answers:

$-3i+\frac{1}{9}$

$\frac{27}{35}i+\frac{3}{35}$

$\frac{27}{82}i-\frac{3}{82}$

$\frac{27}{13}i+\frac{3}{13}$

$\frac{27}{82}i+\frac{3}{82}$

Correct answer:

$\frac{27}{82}i+\frac{3}{82}$

Explanation:

In order to simplify this expression, we will need to multiply the numerator and denominator by the conjugate of the denominator.

$\frac{3i}{i+9}\cdot \frac{i-9}{i-9}$

Simplify both the top and bottom. Recall that: $i=\sqrt{-1}$

3i(i-9) = 3i^2-27i = 3(-1)-27i = -27i-3

(i+9)(i-9) = i^2-81 = (-1)-81 =-82

Divide the numerator with the denominator.

$\frac{ -27i-3}{-82}$

Simplify this term and split the fraction.

The answer is: $\frac{27}{82}i+\frac{3}{82}$

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Example Question #31 : Imaginary Numbers

Solve: $\frac{6i+6}{i-3}$

Possible Answers:

$-\frac{12}{5}i+\frac{6}{5}$

$-\frac{12}{5}i+\frac{3}{5}$

$-\frac{6}{5}i+\frac{6}{5}$

$-\frac{3}{5}i+\frac{3}{5}$

$-\frac{12}{5}i-\frac{6}{5}$

Correct answer:

$-\frac{12}{5}i-\frac{6}{5}$

Explanation:

Multiply the top and bottom of the expression by the conjugate of the denominator.

$\frac{6i+6}{i-3}\times\frac{i+3}{i+3}$

Simplify the top and bottom using the FOIL method.

(6i+6)(i+3) = 6i^2+18i+6i+18 = 6i^2+24i+18

Note that: $i^2= i\cdot i = -1$

Replace the term. The numerator becomes:

6i^2+24i+18 =6(-1)+24i+18 = 24i+12

Simplify the denominator.

(i-3)(i+3)= i^2-9 = (-1)-9 = -10

Divide the numerator with the denominator.

$\frac{ 24i+12}{-10} = -\frac{12}{5}i-\frac{6}{5}$

The answer is: $-\frac{12}{5}i-\frac{6}{5}$

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Example Question #39 : Imaginary Numbers

Simplify: $\frac{3i}{3i-1}$

Possible Answers:

$-\frac{9}{10}i+\frac{3}{10}$

$-\frac{3}{10}i+\frac{1}{9}$

$-\frac{3}{4}i+\frac{3}{5}$

$-\frac{1}{3}i+\frac{9}{10}$

$-\frac{3}{10}i+\frac{9}{10}$

Correct answer:

$-\frac{3}{10}i+\frac{9}{10}$

Explanation:

Multiply the top and bottom of this fraction by the conjugate of the denominator.

$\frac{3i}{3i-1}\cdot \frac{3i+1}{3i+1}$

Simplify the numerator. Recall that $i=\sqrt{-1}$ , and $i^2= i\cdot i = -1$ .

3i(3i+1) = 9i^2+3i = 9(-1)+3i= 3i-9

Simplify the denominator by FOIL method.

(3i-1)(3i+1) = 9i^2-1 = 9(-1)-1=-10

Divide the numerator with the denominator.

$\frac{3i-9}{-10}$

The answer is: $-\frac{3}{10}i+\frac{9}{10}$

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Example Question #31 : Complex Imaginary Numbers

Simplify, if possible: $\frac{5}{i-7}$

Possible Answers:

$-\frac{1}{35}i-\frac{1}{7}$

$-\frac{1}{35}i-\frac{5}{7}$

$\textup{The expression is already simplified.}$

10i-7

$-\frac{1}{10}i-\frac{7}{10}$

Correct answer:

$-\frac{1}{10}i-\frac{7}{10}$

Explanation:

Multiply the top and the bottom of the fraction by the conjugate of the denominator.

$\frac{5}{i-7} \cdot \frac{i+7}{i+7}$

Expand the top and bottom.

5(i+7) = 5i+35

(i-7)(i+7) = i^2-49

Recall that:

$i=\sqrt{-1}$

$i^2= i\cdot i = -1$

Replace the imaginary term for i^2 .

$\frac{5}{i-7} \cdot \frac{i+7}{i+7} = \frac{5i+35}{i^2-49} = \frac{5i+35}{(-1)-49}=\frac{5i+35}{-50}$

Simplify this fraction.

The answer is: $-\frac{1}{10}i-\frac{7}{10}$

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Example Question #41 : Imaginary Numbers

Evaluate: $\frac{2i}{i-3}+\frac{i}{i+1}$

Possible Answers:

$-\frac{1}{10}i+\frac{17}{10}$

$-\frac{7}{10}i+\frac{7}{10}$

$-\frac{3}{20}i-\frac{7}{20}$

$-\frac{1}{20}i+\frac{7}{20}$

$-\frac{1}{10}i+\frac{7}{10}$

Correct answer:

$-\frac{1}{10}i+\frac{7}{10}$

Explanation:

Identify the least common denominator of the imaginary terms by using the FOIL method.

(i-3)(i+1) = (i)(i)+(i)(1)+(-3)(i)+(-3)(1)

Simplify the terms. Recall that:

$i=\sqrt{-1}$ and $i^2= i\cdot i = -1$

The expression becomes:

i^2-2i-3 = (-1)-2i-3 = -2i-4

Convert the fractions of the original problem.

$\frac{2i}{i-3}+\frac{i}{i+1} = \frac{2i(i+1)}{(i-3)(i+1)}+\frac{i(i-3)}{(i+1)(i-3)}$

Simplify the numerators.

$\frac{2i^2+2i}{(i-3)(i+1)}+\frac{i^2-3i}{(i+1)(i-3)} = \frac{3i^2-i}{(i+1)(i-3)}$

Replace the i^2 term.

$= \frac{3(-1)-i}{(i+1)(i-3)} = \frac{-i-3}{(i+1)(i-3)}$

Replace the denominator with the value of the LCD.

$\frac{-i-3}{-2i-4}$

Factor out a negative one.

$\frac{-i-3}{-2i-4} = \frac{-1(i+3)}{-1(2i+4)} = \frac{i+3}{2i+4}$

Simplify by multiplying both the top and bottom by the conjugate of the denominator. Factor both the top and bottom using the FOIL method.

$\frac{i+3}{2i+4}\cdot \frac{2i-4}{2i-4} = \frac{2i^2+2i-12}{4i^2-16}$

Replace the i^2 term.

$\frac{2i^2+2i-12}{4i^2-16}= \frac{2(-1)+2i-12}{4(-1)-16}= \frac{2i-14}{-20}$

Simplify the fraction.

The answer is: $-\frac{1}{10}i+\frac{7}{10}$

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Example Question #42 : Imaginary Numbers

Simplify: $\frac{2i-3}{i-3}$

Possible Answers:

$\frac{3}{10}i-\frac{11}{10}$

$-\frac{3}{10}i+\frac{11}{10}$

$-\frac{3}{5}i+\frac{11}{5}$

$-\frac{5}{8}i+\frac{7}{8}$

$-\frac{3}{8}i+\frac{5}{8}$

Correct answer:

$-\frac{3}{10}i+\frac{11}{10}$

Explanation:

Multiply the top and the bottom by the conjugate of the denominator.

$\frac{2i-3}{i-3} \cdot \frac{i+3}{i+3}$

Since the i-terms are imaginary numbers, write out the actual value of and i^2 .

$i=\sqrt{-1}$

$i^2= i\cdot i = -1$

Simplify both top and bottom by the FOIL method.

(2i-3)(i+3) = 2i^2+3i-9 = 2(-1)+3i-9 = 3i-11

(i-3)(i+3) = i^2-9 = (-1)-9 = -10

Divide the two terms.

$\frac{ 3i-11}{-10}$

The answer is: $-\frac{3}{10}i+\frac{11}{10}$

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Example Question #43 : Imaginary Numbers

Simplify: $-\frac{i}{i-1}$

Possible Answers:

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

$-\frac{2}{3}i+\frac{2}{3}$

$\frac{1}{4}i-\frac{1}{4}$

$-\frac{4}{3}i+\frac{4}{3}$

$-\frac{1}{2}i+\frac{1}{2}$

Correct answer:

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

Explanation:

Multiply the top and bottom by the conjugate of the denominator.

$-\frac{i}{i-1} \cdot \frac{i+1}{i+1}$

Simplify the top and bottom. The bottom can be simplified by the FOIL method.

i(i+1) = i^2+i

(i-1)(i+1) = i^2-1

The expression becomes: $-\frac{i^2+i}{i^2-1}$

Recall that $i=\sqrt{-1}$ and $i^2= i\cdot i = -1$ . Replace the values.

$-\frac{(-1)+i}{(-1)-1} = -\frac{i-1}{-2} = \frac{i-1}{2}$

The answer is: $\frac{1}{2}i-\frac{1}{2}$

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Example Question #44 : Imaginary Numbers

Simplify, if possible: $\frac{1-i}{2-i}$

Possible Answers:

$-\frac{1}{2}i+\frac{3}{2}$

$-\frac{2}{5}i+\frac{4}{5}$

$-\frac{1}{10}i+\frac{3}{10}$

$-\frac{3}{5}i+\frac{3}{5}$

$-\frac{1}{5}i+\frac{3}{5}$

Correct answer:

$-\frac{1}{5}i+\frac{3}{5}$

Explanation:

In order to simplify this, we will need to multiply both the top and bottom by the conjugate of the denominator.

$\frac{1-i}{2-i} \cdot \frac{2+i}{2+i}$

Simplify the top and bottom by FOIL method. Recall that: $i=\sqrt{-1}$ and $i^2= i\cdot i = -1$ .

Start with the numerator.

(1-i)(2+i) = 2-2i+i-i^2 = -i^2-i+2

-i^2-i+2 = -(-1)-i+2 = 1-i+2 = 3-i

Solve the denominator.

(2-i)(2+i) = 4-i^2 = 4-(-1) =5

Divide the numerator and denominator.

$\frac{3-i}{5}$

The answer is: $-\frac{1}{5}i+\frac{3}{5}$

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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #31 : Complex Imaginary Numbers

Example Question #36 : Imaginary Numbers

Example Question #31 : Complex Imaginary Numbers

Example Question #31 : Imaginary Numbers

Example Question #39 : Imaginary Numbers

Example Question #31 : Complex Imaginary Numbers

Example Question #41 : Imaginary Numbers

Example Question #42 : Imaginary Numbers

Example Question #43 : Imaginary Numbers

Example Question #44 : Imaginary Numbers

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