Precalculus Unit 1 Review: Step-by-Step Study Guidance

Q1. Solve the quadratic equation: 2x(x + 2) = 7. Give exact answers, simplifying all radicals and reducing all fractions. Include real and imaginary solutions.

Background

Topic: Quadratic Equations and the Quadratic Formula

This question tests your ability to solve quadratic equations using the quadratic formula, including simplifying radicals and handling both real and complex solutions.

Key Terms and Formulas

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

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

• Discriminant: $b^2 - 4ac$ (determines the nature of the roots)

Step-by-Step Guidance

1. Expand the left side of the equation: $2x(x + 2) = 2x^2 + 4x$.

2. Rewrite the equation in standard quadratic form: $2x^2 + 4x - 7 = 0$.

3. Identify the coefficients: $a = 2$, $b = 4$, $c = -7$.

4. Set up the quadratic formula with these values: $x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 2 \cdot (-7)}}{2 \cdot 2}$.

5. Calculate the discriminant $b^2 - 4ac$ and simplify under the radical, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Final Answer: $x = \frac{-2 \pm 3\sqrt{2}}{2}$

After simplifying the quadratic formula, the solutions are exact and in simplest radical form. Both roots are real because the discriminant is positive.

Q10. Simplify and write the answer in the form $a + bi$: (3 − 2i)(4 + 5i)

Background

Topic: Operations with Complex Numbers

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

Key Terms and Formulas

• Complex Number: $a + bi$, where $i = \sqrt{-1}$

• Multiplying Complex Numbers: Use the distributive property (FOIL method)

• Remember: $i^2 = -1$

Step-by-Step Guidance

1. Apply the distributive property: $(3 - 2i)(4 + 5i)$.

2. Multiply each term: $3 \times 4$, $3 \times 5i$, $-2i \times 4$, $-2i \times 5i$.

3. Combine like terms and use $i^2 = -1$ to simplify the imaginary part.

4. Group the real and imaginary parts together, but do not combine to the final $a + bi$ form yet.

Try solving on your own before revealing the answer!

Final Answer: $22 + 7i$

After multiplying and simplifying, the result is $22 + 7i$.

Q16. Simplify the complex rational expression: $\left(\frac{x-2}{x+2} - 1\right) \div \left(\frac{x-2}{x+2} + 1\right)$

Background

Topic: Simplifying Complex Rational Expressions

This question tests your ability to simplify expressions involving fractions within fractions (complex rational expressions).

Key Terms and Formulas

• Complex Rational Expression: A fraction where the numerator and/or denominator contains fractions.

• To simplify: Combine terms in the numerator and denominator, then divide (multiply by the reciprocal).

Step-by-Step Guidance

1. Rewrite the numerator and denominator as single fractions by expressing 1 as $\frac{x+2}{x+2}$.

2. Simplify the numerator: $\frac{x-2}{x+2} - \frac{x+2}{x+2}$.

3. Simplify the denominator: $\frac{x-2}{x+2} + \frac{x+2}{x+2}$.

4. Write the overall expression as a division of two fractions, and set up the multiplication by the reciprocal.

5. Simplify the resulting expression, but do not reduce to the final answer yet.

Try solving on your own before revealing the answer!

Final Answer: $-\frac{2}{x}$

After simplifying, the complex rational expression reduces to $-\frac{2}{x}$.

Q22. Simplify the expression: $(3x^{-2}y^3)^{-2} \cdot (2xy^{-1})^3$. Express your answer with only positive exponents.

Background

Topic: Exponent Rules and Simplifying Expressions

This question tests your understanding of exponent rules, including negative and zero exponents, and how to simplify expressions with multiple variables.

Key Terms and Formulas

• Power of a Power: $(a^m)^n = a^{mn}$

• Product of Powers: $a^m \cdot a^n = a^{m+n}$

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$

• Zero Exponent: $a^0 = 1$ (if $a \neq 0$)

Step-by-Step Guidance

1. Apply the power rule to each factor: $(3x^{-2}y^3)^{-2}$ and $(2xy^{-1})^3$.

2. Distribute the exponents to each term inside the parentheses.

3. Multiply the resulting expressions together, combining like bases by adding exponents.

4. Rewrite the expression so that all exponents are positive, but do not simplify to the final answer yet.

Try solving on your own before revealing the answer!

Final Answer: $\frac{8x^7}{9y^9}$

After applying exponent rules and simplifying, the expression becomes $\frac{8x^7}{9y^9}$.

Q28. Solve the absolute value inequality: $9|1 + 8n| - 3 \leq 78$

Background

Topic: Absolute Value Equations and Inequalities

This question tests your ability to solve inequalities involving absolute value, including isolating the absolute value and splitting into two cases.

Key Terms and Formulas

• Absolute Value Inequality: $|A| \leq B$ means $-B \leq A \leq B$

• To solve: Isolate the absolute value, then write and solve the compound inequality.

Step-by-Step Guidance

1. Isolate the absolute value term: Add 3 to both sides, then divide by 9.

2. Set up the compound inequality: $|1 + 8n| \leq$ (your result from step 1).

3. Rewrite as two inequalities: $-(\text{value}) \leq 1 + 8n \leq (\text{value})$.

4. Solve for $n$ in the compound inequality, but do not write the final interval yet.

Try solving on your own before revealing the answer!

Final Answer: $-\frac{5}{4} \leq n \leq 1$

The solution is the interval $-\frac{5}{4} \leq n \leq 1$.

Q50. Find the domain of the function: $f(x) = \sqrt{x} - \frac{2}{x^2 + x - 6}$

Background

Topic: Domain of Functions (Radicals and Rational Expressions)

This question tests your ability to find the domain of a function that includes both a square root and a rational expression.

Key Terms and Formulas

• Domain: The set of all real numbers for which the function is defined.

• Square Root: The radicand must be non-negative ($x \geq 0$).

• Rational Expression: The denominator cannot be zero ($x^2 + x - 6 \neq 0$).

Step-by-Step Guidance

1. Set the radicand $x$ greater than or equal to zero: $x \geq 0$.

2. Set the denominator $x^2 + x - 6$ not equal to zero, and factor to find excluded values.

3. Combine the restrictions from both the radical and the denominator to write the domain in interval notation, but do not write the final answer yet.

Try solving on your own before revealing the answer!

Final Answer: $(2, \infty)$

The domain is all real numbers greater than 2: $(2, \infty)$.