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 like terms (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 real parts and imaginary parts separately.

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: $(5 - 11) + (-3i - 4i)$.

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: $a(b + c) = ab + ac$.

Step-by-Step Guidance

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

2. Simplify each term: $-3i \times 6$ and $-3i \times (-i)$.

3. Remember to use $i^2 = -1$ when simplifying $-3i \times (-i)$.

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 for binomial multiplication.

• $i^2 = -1$

Step-by-Step Guidance

1. Multiply the first terms: $3 \times -2$.

2. Multiply the outside terms: $3 \times 4i$.

3. Multiply the inside terms: $-4i \times -2$.

4. Multiply the last terms: $-4i \times 4i$.

5. Combine like terms and use $i^2 = -1$ to 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 ability to multiply conjugate pairs and recognize the result is 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. Multiply the first terms: $\frac{3}{4} \times \frac{3}{4}$.

2. Multiply the outside and inside terms: $\frac{3}{4} \times \frac{1}{2}i$ and $-\frac{1}{2}i \times \frac{3}{4}$.

3. Multiply the last terms: $-\frac{1}{2}i \times \frac{1}{2}i$.

4. Combine like terms and use $i^2 = -1$ to simplify.

Try solving on your own before revealing the answer!

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 the conjugate: $\frac{6}{5 + i} \times \frac{5 - i}{5 - i}$.

2. Expand the numerator: $6 \times (5 - i)$.

3. Expand the denominator: $(5 + i)(5 - i)$.

4. Simplify both numerator and denominator, using $i^2 = -1$.

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 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 $5 + 2i$ is $5 - 2i$.

Step-by-Step Guidance

1. Multiply numerator and denominator by the conjugate: $\frac{2 - i}{5 + 2i} \times \frac{5 - 2i}{5 - 2i}$.

2. Expand the numerator: $(2 - i)(5 - 2i)$.

3. Expand the denominator: $(5 + 2i)(5 - 2i)$.

4. Simplify both numerator and denominator, using $i^2 = -1$.

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

2. Check if the quadratic can be factored easily. If not, set up the quadratic formula.

3. Calculate the discriminant: $b^2 - 4ac$.

4. Set up the formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, but do not compute the final values yet.

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, Quadratic Formula, or Square Root Property

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

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Rewrite the equation in standard form: $6x^2 - x - 5 = 0$.

2. Identify $a = 6$, $b = -1$, $c = -5$.

3. Calculate the discriminant: $b^2 - 4ac$.

4. Set up the quadratic formula for $x$.

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

Background

Topic: Quadratic Equations – Solving by Factoring or Quadratic Formula

This question tests your ability to solve a quadratic equation by moving all terms to one side and factoring or using the quadratic formula.

Key Terms and Formulas:

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

• Factoring: Set equation to zero and factor common terms.

Step-by-Step Guidance

1. Rewrite the equation in standard form: $12x^2 - 25x = 0$.

2. Factor out the common term $x$ if possible.

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

Try solving on your own before revealing the answer!

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

Background

Topic: Quadratic Equations – Solving by Square Root Property

This question tests your ability to solve a quadratic equation in the form $ax^2 + c = 0$ using the square root property.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Rewrite the equation as $9x^2 = 16$.

2. Divide both sides by 9 to isolate $x^2$.

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

Try solving on your own before revealing the answer!

Q6. Solve the quadratic equation: $(x - 2)^2 - 36 = 0$

Background

Topic: Quadratic Equations – Solving by Square Root Property

This question tests your ability to solve a quadratic equation in the form $(x - h)^2 = k$ using the square root property.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Add 36 to both sides to isolate the squared term: $(x - 2)^2 = 36$.

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

3. Solve for $x$ by adding 2 to both sides.

Try solving on your own before revealing the answer!