Unit 8 Quadratic Equations Homework 3 Answer Key

Unit 8 Quadratic Equations Homework 3: A Comprehensive Guide to Solving Problems

This guide provides comprehensive solutions and explanations for Unit 8, Homework 3, focusing on quadratic equations. We'll cover various methods for solving these equations, including factoring, the quadratic formula, and completing the square. Understanding these methods is crucial for mastering quadratic equations and their applications in various fields. We'll break down each problem type, providing step-by-step solutions and helpful tips to improve your problem-solving skills.

Understanding Quadratic Equations

Before we dive into the solutions, let's review the fundamentals of quadratic equations. A quadratic equation is an equation of the form:

ax² + bx + c = 0

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The solutions to this equation are the values of 'x' that make the equation true. These solutions are often called roots, zeros, or x-intercepts.

There are several methods to solve quadratic equations:

1. Factoring

Factoring is a method used to find solutions when the quadratic equation can be easily factored into two linear expressions. For example:

x² + 5x + 6 = 0

This equation can be factored as:

(x + 2)(x + 3) = 0

The solutions are x = -2 and x = -3.

Tips for Factoring: Look for common factors, use the difference of squares formula (a² - b² = (a + b)(a - b)), and practice recognizing factoring patterns.

2. Quadratic Formula

The quadratic formula is a general method that works for all quadratic equations, even those that are difficult or impossible to factor. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Where 'a', 'b', and 'c' are the coefficients of the quadratic equation. The ± symbol indicates that there are usually two solutions.

Tips for using the quadratic formula: Carefully substitute the values of a, b, and c into the formula. Pay close attention to the order of operations, especially when dealing with negative numbers and square roots. Remember to simplify the expression as much as possible.

3. Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial, which can then be easily factored. This method is particularly useful when the equation is not easily factored.

Tips for Completing the Square: Remember to add the same value to both sides of the equation to maintain balance. The value to add is (b/2a)². This method can lead to the derivation of the quadratic formula.

Solving Specific Problems from Unit 8, Homework 3

Let's now tackle some sample problems that could be included in Unit 8, Homework 3, demonstrating the application of these methods. Remember to replace these examples with the actual problems from your homework assignment.

Problem 1: Solving by Factoring

Solve the equation: 2x² + 7x + 3 = 0

Solution:

1. Factor the quadratic expression: This quadratic expression factors to (2x + 1)(x + 3) = 0

2. Set each factor equal to zero: 2x + 1 = 0 or x + 3 = 0

3. Solve for x: x = -1/2 or x = -3

Therefore, the solutions are x = -1/2 and x = -3.

Problem 2: Solving using the Quadratic Formula

Solve the equation: 3x² - 5x - 2 = 0

Solution:

1. Identify a, b, and c: a = 3, b = -5, c = -2

2. Substitute into the quadratic formula:

x = [5 ± √((-5)² - 4 * 3 * -2)] / (2 * 3)

3. Simplify:

x = [5 ± √(25 + 24)] / 6

x = [5 ± √49] / 6

x = [5 ± 7] / 6

4. Solve for the two solutions:

x = (5 + 7) / 6 = 2

x = (5 - 7) / 6 = -1/3

Therefore, the solutions are x = 2 and x = -1/3.

Problem 3: Solving by Completing the Square

Solve the equation: x² + 6x + 5 = 0

Solution:

1. Move the constant to the right side: x² + 6x = -5

2. Find the value to complete the square: (6/2)² = 9

3. Add 9 to both sides: x² + 6x + 9 = -5 + 9

4. Factor the perfect square trinomial: (x + 3)² = 4

5. Take the square root of both sides: x + 3 = ±2

6. Solve for x: x = -3 ± 2

Therefore, the solutions are x = -1 and x = -5.

Problem 4: Dealing with Imaginary Solutions

Solve the equation: x² + 4x + 5 = 0

Solution: Using the quadratic formula (a=1, b=4, c=5):

x = [-4 ± √(16 - 4 * 1 * 5)] / 2

x = [-4 ± √(-4)] / 2

x = [-4 ± 2i] / 2

x = -2 ± i

Here, we encounter imaginary solutions because the discriminant (b² - 4ac) is negative. The solutions are complex numbers involving the imaginary unit 'i', where i² = -1.

Problem 5: Word Problem Application

A rectangular garden has a length that is 3 feet more than its width. The area of the garden is 70 square feet. Find the dimensions of the garden.

Solution:

1. Let x represent the width: The length is x + 3.

2. Set up the equation: x(x + 3) = 70

3. Expand and rearrange: x² + 3x - 70 = 0

4. Solve using factoring: (x + 10)(x - 7) = 0

5. Find the solutions: x = -10 or x = 7

6. Interpret the solution: Since width cannot be negative, the width is 7 feet and the length is 10 feet (7 + 3).

Further Practice and Tips for Success

This guide provides a strong foundation for solving quadratic equations. To solidify your understanding, work through additional problems from your textbook or online resources. Focus on understanding the underlying concepts and choosing the most efficient method for solving each equation. Regular practice is key to mastering these techniques. Remember to always check your answers by substituting them back into the original equation.

By mastering the methods outlined in this comprehensive guide, you'll be well-equipped to tackle any quadratic equation problem you encounter in Unit 8, Homework 3, and beyond. Remember to utilize the different methods based on the characteristics of each problem for optimized efficiency. Good luck!