Comprehensive Step-by-Step Guidance for College Algebra Final Exam Review

Q1a. Simplify the complex numbers: (3 − 4i) + (−2 + 7i)

Background

Topic: Complex Numbers (Addition)

This question tests your ability to add complex numbers by combining like terms (real and imaginary parts).

Key Terms and Formulas:

• A complex number is of the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

• To add complex numbers: $(a + bi) + (c + di) = (a + c) + (b + d)i$

Step-by-Step Guidance

1. Identify the real and imaginary parts in each complex number: $3 - 4i$ and $-2 + 7i$.

2. Add the real parts together: $3 + (-2)$.

3. Add the imaginary parts together: $-4i + 7i$.

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Q1b. Simplify the complex numbers: (2 + i)(3 − 2i)

Background

Topic: Complex Numbers (Multiplication)

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

Key Terms and Formulas:

• To multiply: $(a + bi)(c + di) = ac + adi + bci + bdi^2$

• Recall that $i^2 = -1$

Step-by-Step Guidance

1. Apply the distributive property (FOIL): Multiply each term in the first complex number by each term in the second.

2. Compute: $2 \times 3$, $2 \times (-2i)$, $i \times 3$, $i \times (-2i)$.

3. Combine like terms and remember to replace $i^2$ with $-1$.

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Q2a. Find the quotient: $\frac{6 + 2i}{1 + 2i}$

Background

Topic: Division of Complex Numbers

This question tests your ability to divide complex numbers by multiplying numerator and denominator by the conjugate of the denominator.

Key Terms and Formulas:

• Conjugate: For $a + bi$, the conjugate is $a - bi$.

• To divide: $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$

• Remember $i^2 = -1$

Step-by-Step Guidance

1. Identify the conjugate of the denominator: $1 - 2i$.

2. Multiply numerator and denominator by this conjugate.

3. Expand both numerator and denominator using distributive property.

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

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Q3a. Simplify $i^{62}$

Background

Topic: Powers of the Imaginary Unit

This question tests your understanding of the cyclical nature of powers of $i$.

Key Terms and Formulas:

• $i^1 = i$

• $i^2 = -1$

• $i^3 = -i$

• $i^4 = 1$ (and repeats every 4 powers)

Step-by-Step Guidance

1. Divide the exponent (62) by 4 to find the remainder.

2. The value of $i^{62}$ will be the same as $i^r$, where $r$ is the remainder.

3. Use the pattern above to determine the simplified value.

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Q5a. Solve the quadratic equation using factoring: $9x^2 - 12x + 4 = 0$

Background

Topic: Quadratic Equations (Factoring)

This question tests your ability to solve quadratic equations by factoring and applying the zero product property.

Key Terms and Formulas:

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

• Zero product property: If $ab = 0$, then $a = 0$ or $b = 0$

Step-by-Step Guidance

1. Look for a common factor or factor the quadratic expression $9x^2 - 12x + 4$.

2. Set each factor equal to zero.

3. Solve each resulting equation for $x$.

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Q7a. Solve the quadratic equation using the quadratic formula: $x^2 - 3x - 2 = 0$

Background

Topic: Quadratic Formula

This question tests your ability to use the quadratic formula to solve any quadratic equation.

Key Terms and Formulas:

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

• For $ax^2 + bx + c = 0$

Step-by-Step Guidance

1. Identify $a$, $b$, and $c$ from the equation: $a = 1$, $b = -3$, $c = -2$.

2. Plug these values into the quadratic formula.

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

4. Set up the expression for $x$ using the quadratic formula, but do not simplify fully yet.

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Q10a. Solve the inequality: $x^2 - x - 12 < 0$

Background

Topic: Quadratic Inequalities

This question tests your ability to solve quadratic inequalities by finding the zeros and analyzing intervals.

Key Terms and Formulas:

• Quadratic inequality: $ax^2 + bx + c < 0$

• Find zeros by factoring or using the quadratic formula.

• Test intervals between zeros to determine where the inequality holds.

Step-by-Step Guidance

1. Set $x^2 - x - 12 = 0$ and solve for $x$ to find the critical points.

2. These points divide the number line into intervals.

3. Test a value from each interval in the original inequality to see where it is true.

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Q13a. Write the quadratic $y = x^2 - 2x + 3$ in vertex form and find the vertex.

Background

Topic: Completing the Square / Vertex Form of a Parabola

This question tests your ability to rewrite a quadratic in vertex form and identify the vertex.

Key Terms and Formulas:

• Vertex form: $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

• Complete the square to rewrite the quadratic in vertex form.

Step-by-Step Guidance

1. Start with $y = x^2 - 2x + 3$.

2. Group the $x$ terms: $y = (x^2 - 2x) + 3$.

3. Find the value to complete the square: Take half of the coefficient of $x$, square it, and add and subtract it inside the parentheses.

4. Rewrite the expression as a perfect square trinomial plus the constant terms.

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