Step-by-Step Guidance: Quadratic Functions and Complex Numbers (College Algebra)

Q1. Simplify the following expression and write the answer in standard form: (i^{105} - i^{103}) / i^{85}

Background

Topic: Powers of the Imaginary Unit (i)

This question tests your understanding of how to simplify powers of the imaginary unit $i$, and how to write complex numbers in standard form ($a + bi$).

Key Terms and Formulas

• $i$ is the imaginary unit, where $i^2 = -1$.

• Powers of $i$ repeat every 4: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, then the cycle repeats.

• Standard form for a complex number: $a + bi$.

Step-by-Step Guidance

1. Find the remainder when each exponent (105, 103, 85) is divided by 4 to determine the equivalent lower power of $i$ for each.

2. Rewrite $i^{105}$, $i^{103}$, and $i^{85}$ using their equivalent values from the $i$ cycle.

3. Subtract the two numerators and then divide by the denominator, simplifying as much as possible.

4. Express your answer in standard form ($a + bi$).

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Q2. Simplify the following expression and write the answer in standard form: \frac{3 + i}{3 + 5i} + \frac{3 - i}{3 - 5i}

Background

Topic: Operations with Complex Numbers (Addition, Division)

This question tests your ability to add and divide complex numbers, and to express the result in standard form.

Key Terms and Formulas

• To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator.

• Conjugate of $a + bi$ is $a - bi$.

• Standard form: $a + bi$.

Step-by-Step Guidance

1. For each fraction, multiply numerator and denominator by the conjugate of the denominator to rationalize it.

2. Simplify the numerators and denominators using the distributive property and $i^2 = -1$.

3. Add the two resulting fractions together, combining like terms.

4. Express the final result in standard form ($a + bi$).

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Q3. Write the following expression in standard form: (2 - i)^3

Background

Topic: Expanding Complex Numbers (Binomial Theorem)

This question tests your ability to expand a binomial expression with complex numbers and write the result in standard form.

Key Terms and Formulas

• Binomial expansion: $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$.

• Remember $i^2 = -1$ and $i^3 = -i$.

• Standard form: $a + bi$.

Step-by-Step Guidance

1. Expand $(2 - i)^3$ using the binomial theorem or by multiplying $(2 - i)$ by itself three times.

2. Simplify each term, substituting $i^2 = -1$ and $i^3 = -i$ as needed.

3. Combine like terms to write the result in the form $a + bi$.

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Q4. In the given expression, determine the product: \sqrt{-11} \times \sqrt{-7}

Background

Topic: Multiplying Complex Numbers (Square Roots of Negative Numbers)

This question tests your understanding of how to multiply square roots of negative numbers using the imaginary unit $i$.

Key Terms and Formulas

• $\sqrt{-a} = i\sqrt{a}$ for $a > 0$.

• Multiplying two imaginary numbers: $i^2 = -1$.

Step-by-Step Guidance

1. Rewrite each square root in terms of $i$ (e.g., $\sqrt{-11} = i\sqrt{11}$).

2. Multiply the two expressions together, using the properties of $i$.

3. Simplify the result, combining the $i$ terms and the square roots.

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Q5. Perform multiplication in the given expression. Use the standard form to write the final answer: (8 - i)(8 + i)(2 - 6i)

Background

Topic: Multiplying Complex Numbers

This question tests your ability to multiply several complex numbers and express the result in standard form.

Key Terms and Formulas

• Use the distributive property (FOIL) to multiply complex numbers.

• Remember $i^2 = -1$.

• Standard form: $a + bi$.

Step-by-Step Guidance

1. First, multiply $(8 - i)$ and $(8 + i)$ using the difference of squares formula.

2. Multiply the result by $(2 - 6i)$ using the distributive property.

3. Simplify all terms, substituting $i^2 = -1$ where needed.

4. Combine like terms to write the result in standard form.

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Q6. Perform division and express the quotient in standard form: \frac{119}{-i}

Background

Topic: Dividing by Imaginary Numbers

This question tests your ability to divide a real number by an imaginary number and express the result in standard form.

Key Terms and Formulas

• To divide by $i$, multiply numerator and denominator by $i$ to rationalize the denominator.

• Remember $i^2 = -1$.

Step-by-Step Guidance

1. Multiply numerator and denominator by $i$ to eliminate $i$ from the denominator.

2. Simplify the denominator using $i^2 = -1$.

3. Write the result in standard form ($a + bi$).

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Q7. Enlist all values of x that will make the given expression undefined: \frac{5}{x^2 - 7x + 6}

Background

Topic: Domain of Rational Expressions

This question tests your ability to find values that make a rational expression undefined (i.e., where the denominator is zero).

Key Terms and Formulas

• A rational expression is undefined when its denominator equals zero.

• Set the denominator equal to zero and solve for $x$.

Step-by-Step Guidance

1. Set the denominator $x^2 - 7x + 6$ equal to zero.

2. Factor the quadratic equation.

3. Solve for the values of $x$ that make the denominator zero.

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Q8. For the following equation, calculate the discriminant, as well as the number and type of solutions: x^2 - 11x - 12 = 0

Background

Topic: Discriminant of a Quadratic Equation

This question tests your ability to calculate the discriminant and interpret the number and type of solutions for a quadratic equation.

Key Terms and Formulas

• Quadratic equation: $ax^2 + bx + c = 0$.

• Discriminant: $D = b^2 - 4ac$.

• If $D > 0$, two real solutions; if $D = 0$, one real solution; if $D < 0$, no real solutions (complex solutions).

Step-by-Step Guidance

1. Identify $a$, $b$, and $c$ from the equation $x^2 - 11x - 12 = 0$.

2. Plug these values into the discriminant formula $D = b^2 - 4ac$.

3. Calculate the value of $D$.

4. Interpret the value of $D$ to determine the number and type of solutions.

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Q9. Identify the term to be added on the given binomial that will result in a perfect square trinomial. After identifying the term, factor the perfect square trinomial: x^2 - 9x

Background

Topic: Completing the Square

This question tests your ability to complete the square and factor a quadratic expression.

Key Terms and Formulas

• To complete the square for $x^2 + bx$, add $\left(\frac{b}{2}\right)^2$.

• Perfect square trinomial: $x^2 + bx + \left(\frac{b}{2}\right)^2 = (x + \frac{b}{2})^2$.

Step-by-Step Guidance

1. Identify the coefficient $b$ in $x^2 - 9x$.

2. Calculate $\left(\frac{b}{2}\right)^2$ and add it to the expression.

3. Write the resulting trinomial as a perfect square.

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Q10. Find the solutions to the following equation: k^2 - 361 = 0

Background

Topic: Solving Quadratic Equations by Square Roots

This question tests your ability to solve a quadratic equation using the square root property.

Key Terms and Formulas

• Square root property: If $x^2 = a$, then $x = \pm \sqrt{a}$.

Step-by-Step Guidance

1. Rewrite the equation as $k^2 = 361$.

2. Take the square root of both sides, remembering to include both the positive and negative roots.

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Q11. Use the square root rule to solve the given quadratic equation: (-3x + 7)^2 = -135

Background

Topic: Solving Quadratic Equations with Complex Solutions

This question tests your ability to use the square root property to solve a quadratic equation that has complex solutions.

Key Terms and Formulas

• Square root property: If $x^2 = a$, then $x = \pm \sqrt{a}$.

• If $a$ is negative, $\sqrt{a} = i\sqrt{|a|}$.

Step-by-Step Guidance

1. Take the square root of both sides of the equation, remembering to use $i$ for the square root of a negative number.

2. Solve for $x$ by isolating it on one side of the equation.

3. Express your solutions in standard form, showing both the plus and minus cases.

Try solving on your own before revealing the answer!