Unit 8 Quadratic Equations Homework 1 Answer Key

Unit 8 Quadratic Equations Homework 1: A Comprehensive Guide to Solving Quadratic Equations

This comprehensive guide delves into the solutions for a typical Unit 8 Quadratic Equations Homework 1 assignment. We will cover various methods for solving quadratic equations, including factoring, completing the square, the quadratic formula, and graphing, providing detailed explanations and examples for each. This guide aims to not only provide answers but also to solidify your understanding of quadratic equations and their applications.

Understanding Quadratic Equations

Before diving into the solutions, let's review the fundamentals. 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, also known as the roots, zeros, or x-intercepts, represent the x-values where the graph of the quadratic function intersects the x-axis.

Methods for Solving Quadratic Equations

Several methods exist for solving quadratic equations. The best method often depends on the specific equation's characteristics.

1. Factoring:

Factoring is the most straightforward method when the quadratic expression can be easily factored. It involves rewriting the quadratic equation as a product of two linear expressions. For example:

x² + 5x + 6 = 0

This can be factored as:

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

Therefore, the solutions are x = -2 and x = -3. This is because if either (x + 2) or (x + 3) equals zero, the entire equation is satisfied.

Example: Solve x² - 7x + 12 = 0 by factoring.

• Solution: The equation factors to (x - 3)(x - 4) = 0. Therefore, the solutions are x = 3 and x = 4.

2. Completing the Square:

Completing the square is a useful method when factoring is not readily apparent. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored. The general steps are:

• Move the constant term to the right side of the equation.
• Divide the coefficient of the x-term by 2, square it, and add it to both sides of the equation.
• Factor the perfect square trinomial on the left side.
• Take the square root of both sides and solve for x.

Example: Solve x² + 6x + 2 = 0 by completing the square.

• Solution:
  1. x² + 6x = -2
  2. (6/2)² = 9. Add 9 to both sides: x² + 6x + 9 = 7
  3. Factor: (x + 3)² = 7
  4. Take the square root: x + 3 = ±√7
  5. Solve for x: x = -3 ± √7

3. Quadratic Formula:

The quadratic formula is a universal method that works for all quadratic equations, regardless of whether they can be factored easily. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

where a, b, and c are the coefficients from the standard quadratic equation ax² + bx + c = 0.

Example: Solve 2x² - 5x + 2 = 0 using the quadratic formula.

• Solution: Here, a = 2, b = -5, and c = 2. Substituting into the formula: x = [5 ± √((-5)² - 4 * 2 * 2)] / (2 * 2) x = [5 ± √9] / 4 x = [5 ± 3] / 4 Therefore, x = 2 or x = 1/2

4. Graphing:

Graphing a quadratic equation provides a visual representation of its solutions. The x-intercepts of the parabola (the graph of a quadratic equation) represent the solutions to the equation. While graphing can be an effective method for estimating solutions, it is often less precise than algebraic methods for finding exact values. Using graphing calculators or software can significantly improve the accuracy of this method.

Interpreting the Discriminant (b² - 4ac):

The expression inside the square root in the quadratic formula, (b² - 4ac), is called the discriminant. It provides information about the nature of the solutions:

• b² - 4ac > 0: Two distinct real solutions (two x-intercepts).
• b² - 4ac = 0: One real solution (one x-intercept, the vertex touches the x-axis).
• b² - 4ac < 0: No real solutions (the parabola does not intersect the x-axis). The solutions are complex numbers.

Sample Problems and Solutions (Unit 8 Quadratic Equations Homework 1)

Let's assume your homework includes a variety of problems utilizing the methods described above. Here are some sample problems and their solutions:

Problem 1: Solve x² - 9 = 0

• Solution: This can be solved by factoring as a difference of squares: (x - 3)(x + 3) = 0. Therefore, x = 3 or x = -3.

Problem 2: Solve 3x² + 12x = 0

• Solution: Factor out 3x: 3x(x + 4) = 0. Therefore, x = 0 or x = -4.

Problem 3: Solve x² + 4x + 1 = 0 using the quadratic formula.

• Solution: a = 1, b = 4, c = 1. Using the quadratic formula: x = [-4 ± √(4² - 4 * 1 * 1)] / (2 * 1) x = [-4 ± √12] / 2 x = -2 ± √3

Problem 4: Solve x² - 6x + 9 = 0 by completing the square.

• Solution:
  1. x² - 6x = -9
  2. (-6/2)² = 9. Add 9 to both sides: x² - 6x + 9 = 0
  3. Factor: (x - 3)² = 0
  4. Take the square root: x - 3 = 0
  5. Solve for x: x = 3

Problem 5: Determine the nature of the roots of the equation 2x² + 3x + 5 = 0 without solving.

• Solution: Calculate the discriminant: b² - 4ac = 3² - 4 * 2 * 5 = 9 - 40 = -31. Since the discriminant is negative, there are no real solutions; the solutions are complex numbers.

Problem 6 (Word Problem): The area of a rectangular garden is 12 square meters. If the length is 3 meters longer than the width, find the dimensions of the garden.

• Solution: Let w represent the width. The length is w + 3. The area is w(w + 3) = 12. This leads to the quadratic equation w² + 3w - 12 = 0. Factoring gives (w - 3)(w + 4) = 0. Since width cannot be negative, the width is 3 meters, and the length is 6 meters.

Advanced Concepts and Further Exploration

This guide provides a strong foundation for understanding and solving quadratic equations. However, further exploration into more advanced concepts can deepen your comprehension. These include:

• Solving quadratic inequalities: These involve finding the range of x-values that satisfy an inequality involving a quadratic expression.
• Complex numbers: Understanding complex numbers is crucial when dealing with quadratic equations that have no real solutions.
• Applications of quadratic equations: Exploring real-world applications, such as projectile motion, optimization problems, and area calculations, reinforces the practical significance of quadratic equations.

By mastering these techniques and exploring these further concepts, you'll build a strong foundation in algebra and prepare yourself for more advanced mathematical studies. Remember, consistent practice is key to solidifying your understanding and improving your problem-solving skills. Don't hesitate to review the concepts and work through additional problems to further enhance your proficiency.