College Algebra Final Review – Step-by-Step Study Guidance

Q1. Complete the following complex number operations. Write your answer in standard form a + bi.

Background

Topic: Complex Numbers

This question tests your ability to perform arithmetic operations with complex numbers and express the result in the standard form $a + bi$.

Key Terms and Formulas:

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

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

Step-by-Step Guidance

1. Identify the operation(s) required (addition, subtraction, multiplication, or division) between the given complex numbers.

2. Apply the operation, combining like terms (real with real, imaginary with imaginary).

3. If multiplication is involved, use the distributive property and remember to replace $i^2$ with $-1$.

4. Write the result in the form $a + bi$.

Try solving on your own before revealing the answer!

Q2. Simplify $i^{50}$

Background

Topic: Powers of the Imaginary Unit

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

Key Terms and Formulas:

• $i^1 = i$

• $i^2 = -1$

• $i^3 = -i$

• $i^4 = 1$

• Powers of $i$ repeat every 4: $i^n = i^{n \mod 4}$

Step-by-Step Guidance

1. Divide the exponent (50) by 4 to find the remainder.

2. Use the remainder to determine which value in the cycle $i, -1, -i, 1$ corresponds to $i^{50}$.

3. Express your answer as $1$,$i$, $-1$, or$-i$ as appropriate.

Try solving on your own before revealing the answer!

Q3. Solve the quadratic by completing the square: $x^2 + 5x - 36 = 0$

Background

Topic: Quadratic Equations – Completing the Square

This question tests your ability to solve a quadratic equation by rewriting it in a perfect square form.

Key Terms and Formulas:

• Completing the square: $x^2 + bx + c = 0$ can be rewritten as $(x + \frac{b}{2})^2 =$ (some value)

Step-by-Step Guidance

1. Move the constant term to the other side: $x^2 + 5x = 36$.

2. Take half of the coefficient of $x$ (which is 5), square it, and add it to both sides: $\left(\frac{5}{2}\right)^2$.

3. Rewrite the left side as a perfect square trinomial: $(x + \frac{5}{2})^2$.

4. Set the equation equal to the new value on the right side and prepare to solve for $x$ by taking square roots.

Try solving on your own before revealing the answer!

Q4. Solve the quadratic using the quadratic formula: $x^2 = 6x - 45$

Background

Topic: Quadratic Equations – Quadratic Formula

This question tests your ability to use the quadratic formula to 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: $x^2 - 6x + 45 = 0$.

2. Identify $a$, $b$, and $c$ from the equation.

3. Plug these values into the quadratic formula.

4. Simplify under the square root (the discriminant) and set up the expressions for $x$.

Try solving on your own before revealing the answer!

Q5. Solve the following inequalities. Write your answer in interval notation: $8 < 2 + 2x < 10$

Background

Topic: Compound Inequalities

This question tests your ability to solve compound inequalities and express the solution in interval notation.

Key Terms and Formulas:

• Compound inequality: An inequality with two comparison symbols, solved by isolating the variable in the middle.

Step-by-Step Guidance

1. Subtract 2 from all three parts of the inequality.

2. Divide all three parts by 2 to solve for $x$.

3. Write the solution as an interval.

Try solving on your own before revealing the answer!

Q6. Solve the following absolute value equation: $|6 - 5x| = 2$

Background

Topic: Absolute Value Equations

This question tests your ability to solve equations involving absolute values.

Key Terms and Formulas:

• For $|A| = k$, $A = k$ or $A = -k$

Step-by-Step Guidance

1. Set up two equations: $6 - 5x = 2$ and $6 - 5x = -2$.

2. Solve each equation for $x$.

3. Write both solutions.

Try solving on your own before revealing the answer!

Q7. Solve the following absolute value inequality: $|2x + 1| < 11$

Background

Topic: Absolute Value Inequalities

This question tests your ability to solve inequalities involving absolute values.

Key Terms and Formulas:

• For $|A| < k$, $-k < A < k$

Step-by-Step Guidance

1. Rewrite the inequality as $-11 < 2x + 1 < 11$.

2. Subtract 1 from all parts.

3. Divide all parts by 2 to solve for $x$.

4. Express the solution in interval notation.

Try solving on your own before revealing the answer!

Q8. Find the x-intercept of the graph $7x + 2y = 14$

Background

Topic: Linear Equations – Intercepts

This question tests your ability to find the x-intercept of a linear equation.

Key Terms and Formulas:

• x-intercept: The point where $y = 0$.

Step-by-Step Guidance

1. Set $y = 0$ in the equation $7x + 2y = 14$.

2. Solve for $x$.

3. Write the x-intercept as an ordered pair $(x, 0)$.

Try solving on your own before revealing the answer!

Q9. The graph of $(x - 3)^2 + (y + 5)^2 = 16$ has center with coordinates ___

Background

Topic: Circles – Standard Form

This question tests your ability to identify the center of a circle from its equation.

Key Terms and Formulas:

• Standard form: $(x - h)^2 + (y - k)^2 = r^2$

• Center: $(h, k)$

Step-by-Step Guidance

1. Compare the given equation to the standard form.

2. Identify $h$ and $k$ (remember the signs inside the parentheses).

3. Write the center as an ordered pair.

Try solving on your own before revealing the answer!