College Algebra Study Guide: Quadratics, Complex Numbers, and Equations

Q1. Solve the equation by the zero-factor property: $2x^2 = -24x - 64$

Background

Topic: Quadratic Equations and Zero-Factor Property

This question tests your ability to solve a quadratic equation by first rewriting it in standard form and then factoring to find the solutions.

Key Terms and Formulas:

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

• Zero-Factor Property: If $ab = 0$, then $a = 0$ or $b = 0$.

Step-by-Step Guidance

1. Rewrite the equation in standard form by moving all terms to one side: $2x^2 + 24x + 64 = 0$.

2. Factor out the greatest common factor if possible.

3. Factor the resulting quadratic expression completely.

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

Try solving on your own before revealing the answer!

Q2. Simplify the power of $i$: $\frac{1}{i^{27}}$

Background

Topic: Powers of the Imaginary Unit $i$

This question tests your understanding of the cyclical nature of powers of $i$ and how to simplify expressions involving them.

Key Terms and Formulas:

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

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

Step-by-Step Guidance

1. Find the remainder when the exponent (27) is divided by 4 to determine the equivalent lower power of $i$.

2. Rewrite $i^{27}$ as $i^r$ where $r$ is the remainder from step 1.

3. Recall the values for $i^1, i^2, i^3, i^4$ and substitute accordingly.

4. Take the reciprocal since the expression is $\frac{1}{i^{27}}$.

Try solving on your own before revealing the answer!

Q3. Solve the formula for the indicated variable: $A = bh$, for $h$

Background

Topic: Literal Equations and Solving for a Variable

This question tests your ability to rearrange a formula to solve for a specific variable.

Key Terms and Formulas:

• Literal Equation: An equation involving two or more variables.

• To solve for a variable, isolate it on one side of the equation.

Step-by-Step Guidance

1. Start with the equation $A = bh$.

2. To solve for $h$, divide both sides of the equation by $b$ (assuming $b \neq 0$).

Try solving on your own before revealing the answer!

Q4. Decide whether or not the equations are equivalent: $3x - 2 = 8$ and $-3x + 8 = -2$

Background

Topic: Equivalent Equations

This question tests your understanding of when two equations have the same solution set.

Key Terms and Formulas:

• Equivalent Equations: Two equations that have exactly the same solutions.

Step-by-Step Guidance

1. Solve the first equation for $x$.

2. Solve the second equation for $x$.

3. Compare the solutions to determine if they are the same.

Try solving on your own before revealing the answer!

Q5. Solve the formula for the indicated variable: $S = 2\pi rh + 2\pi r^2$, for $h$

Background

Topic: Literal Equations and Solving for a Variable

This question tests your ability to manipulate a formula to isolate a specific variable.

Key Terms and Formulas:

• Surface area of a cylinder: $S = 2\pi rh + 2\pi r^2$

• To solve for $h$, isolate the term containing $h$ and then divide both sides by the coefficient of $h$.

Step-by-Step Guidance

1. Subtract $2\pi r^2$ from both sides to isolate the $2\pi rh$ term.

2. Divide both sides by $2\pi r$ to solve for $h$.

Try solving on your own before revealing the answer!