College Algebra Exam 1 Review – Step-by-Step Study Guidance

Q1a. Simplify: (5 - 3i) - (11 + 4i)

Background

Topic: Complex Numbers – Addition and Subtraction

This question tests your ability to add and subtract complex numbers by combining real and imaginary parts.

Key Terms and Formulas:

• Complex number: $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

• To add or subtract complex numbers, combine like terms (real with real, imaginary with imaginary).

Step-by-Step Guidance

1. Write both complex numbers in standard form: $(5 - 3i)$ and $(11 + 4i)$.

2. Distribute the negative sign to the second complex number: $(5 - 3i) - 11 - 4i$.

3. Group the real parts and the imaginary parts separately.

4. Combine the real parts and combine the imaginary parts.

Try solving on your own before revealing the answer!

Q1b. Simplify: -3i(6 - i)

Background

Topic: Complex Numbers – Multiplication

This question tests your ability to multiply a complex number by a scalar and another imaginary number.

Key Terms and Formulas:

• Recall that $i^2 = -1$.

• Use the distributive property to multiply.

Step-by-Step Guidance

1. Apply the distributive property: $-3i \times 6$ and $-3i \times (-i)$.

2. Simplify each term, remembering that $i \times i = i^2$.

3. Replace $i^2$ with $-1$ and simplify further.

Try solving on your own before revealing the answer!

Q1c. Simplify: (3 - 4i)(-2 + 4i)

Background

Topic: Complex Numbers – Multiplication (FOIL Method)

This question tests your ability to multiply two complex numbers using the distributive (FOIL) method.

Key Terms and Formulas:

• FOIL: First, Outside, Inside, Last terms.

• $i^2 = -1$.

Step-by-Step Guidance

1. Multiply each term in the first parenthesis by each term in the second parenthesis (FOIL).

2. Combine like terms (real and imaginary parts).

3. Replace $i^2$ with $-1$ and simplify.

Try solving on your own before revealing the answer!

Q1d. Simplify: (\frac{3}{4} - \frac{1}{2}i)(\frac{3}{4} + \frac{1}{2}i)

Background

Topic: Complex Numbers – Multiplication of Conjugates

This question tests your understanding of multiplying a complex number by its conjugate, which results in a real number.

Key Terms and Formulas:

• Conjugate: $a + bi$ and $a - bi$.

• $(a + bi)(a - bi) = a^2 + b^2$ (since $i^2 = -1$).

Step-by-Step Guidance

1. Apply the formula for multiplying conjugates.

2. Square the real part and the imaginary part (without $i$).

3. Add the results, remembering $i^2 = -1$.

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Q1e. Simplify: \frac{6}{5 + i}

Background

Topic: Complex Numbers – Rationalizing the Denominator

This question tests your ability to write a complex fraction in standard form by rationalizing the denominator.

Key Terms and Formulas:

• To rationalize, multiply numerator and denominator by the conjugate of the denominator.

• Conjugate of $5 + i$ is $5 - i$.

Step-by-Step Guidance

1. Multiply numerator and denominator by $5 - i$.

2. Expand both numerator and denominator using distributive property.

3. In the denominator, use $(a + b)(a - b) = a^2 - b^2$ and $i^2 = -1$.

4. Simplify the resulting expression to standard form $a + bi$.

Try solving on your own before revealing the answer!

Q1f. Simplify: \frac{2 - i}{5 + 2i}

Background

Topic: Complex Numbers – Rationalizing the Denominator

This question tests your ability to express a complex fraction in standard form by rationalizing the denominator.

Key Terms and Formulas:

• Multiply numerator and denominator by the conjugate of the denominator ($5 - 2i$).

• Use distributive property and $i^2 = -1$.

Step-by-Step Guidance

1. Multiply numerator and denominator by $5 - 2i$.

2. Expand both numerator and denominator.

3. In the denominator, use $(a + bi)(a - bi) = a^2 + b^2$.

4. Simplify to standard form $a + bi$.

Try solving on your own before revealing the answer!

Q2. Solve the quadratic equation: $x^2 - 5x - 14 = 0$

Background

Topic: Quadratic Equations – Solving by Factoring, Quadratic Formula, or Square Root Property

This question tests your ability to solve a quadratic equation using various methods.

Key Terms and Formulas:

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Factoring: Express as $(x - p)(x - q) = 0$

• Square root property: Used when equation is in the form $(x - h)^2 = k$

Step-by-Step Guidance

1. Identify $a$, $b$, and $c$ in the equation $x^2 - 5x - 14 = 0$.

2. Decide which method to use (factoring, quadratic formula, or square root property).

3. If factoring, look for two numbers that multiply to $-14$ and add to $-5$.

4. If using the quadratic formula, substitute $a$, $b$, and $c$ into the formula.

Try solving on your own before revealing the answer!

Q3. Solve the quadratic equation: $6x^2 - x = 5$

Background

Topic: Quadratic Equations – Solving by Factoring or Quadratic Formula

This question tests your ability to rearrange and solve a quadratic equation.

Key Terms and Formulas:

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Factoring: Set equation to zero and factor.

Step-by-Step Guidance

1. Move all terms to one side to set the equation to zero: $6x^2 - x - 5 = 0$.

2. Identify $a$, $b$, and $c$.

3. Attempt to factor or use the quadratic formula.

4. Set up the formula or factors, but do not solve completely yet.

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Q4. Solve the quadratic equation: $12x^2 = 25x$

Background

Topic: Quadratic Equations – Solving by Factoring

This question tests your ability to rearrange and factor a quadratic equation.

Key Terms and Formulas:

• Set the equation to zero: $12x^2 - 25x = 0$.

• Factor out common terms.

Step-by-Step Guidance

1. Move all terms to one side: $12x^2 - 25x = 0$.

2. Factor out $x$.

3. Set each factor equal to zero and solve for $x$.

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Q5. Solve the quadratic equation: $9x^2 - 16 = 0$

Background

Topic: Quadratic Equations – Difference of Squares

This question tests your ability to recognize and solve a difference of squares equation.

Key Terms and Formulas:

• Difference of squares: $a^2 - b^2 = (a - b)(a + b)$

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

Step-by-Step Guidance

1. Recognize the equation as a difference of squares.

2. Factor as $(3x - 4)(3x + 4) = 0$ or use the square root property.

3. Solve for $x$ by setting each factor equal to zero.

Try solving on your own before revealing the answer!