Unit 4 Solving Quadratic Equations Answer Key

Solving quadratic equations is a fundamental skill in algebra with far-reaching applications in various fields, from physics and engineering to economics and computer science. Mastering the techniques to find the solutions, also known as roots or zeros, of quadratic equations is crucial for anyone pursuing further studies in mathematics or related disciplines. This article provides a comprehensive exploration of methods for solving quadratic equations, complete with illustrative examples and a detailed "answer key" approach, ensuring a thorough understanding of the subject.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is:

ax² + bx + c = 0

Where:

• a, b, and c are constants, with a ≠ 0
• x represents the unknown variable

The solutions to this equation are the values of x that satisfy the equation. These solutions can be real or complex numbers.

Methods for Solving Quadratic Equations

There are several methods to solve quadratic equations, each with its advantages and suitability depending on the specific equation:

1. Factoring
2. Completing the Square
3. Quadratic Formula

1. Factoring

Factoring involves expressing the quadratic equation as a product of two linear factors. This method is efficient when the quadratic expression can be easily factored.

Steps for Solving by Factoring:

1. Rearrange the equation: Ensure the equation is in the standard form ax² + bx + c = 0.
2. Factor the quadratic expression: Find two binomials that multiply to give the quadratic expression.
3. Set each factor equal to zero: If the product of two factors is zero, then at least one of the factors must be zero.
4. Solve for x: Solve each linear equation to find the values of x.

Example 1:

Solve the quadratic equation: x² - 5x + 6 = 0

1. The equation is already in standard form.
2. Factor the quadratic expression: x² - 5x + 6 = (x - 2)(x - 3)
3. Set each factor equal to zero:
   • x - 2 = 0
   • x - 3 = 0
4. Solve for x:
   • x = 2
   • x = 3

Thus, the solutions are x = 2 and x = 3.

Example 2:

Solve the quadratic equation: 2x² + 7x + 3 = 0

1. The equation is already in standard form.
2. Factor the quadratic expression: 2x² + 7x + 3 = (2x + 1)(x + 3)
3. Set each factor equal to zero:
   • 2x + 1 = 0
   • x + 3 = 0
4. Solve for x:
   • 2x = -1 → x = -1/2
   • x = -3

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

2. Completing the Square

Completing the square is a method that transforms the quadratic equation into a perfect square trinomial, making it easier to solve. This method is particularly useful when the quadratic expression is not easily factorable.

Steps for Solving by Completing the Square:

1. Rearrange the equation: Rewrite the equation in the form ax² + bx = -c.
2. Divide by a: If a ≠ 1, divide the entire equation by a.
3. Complete the square: Add (b/2a)² to both sides of the equation. This makes the left side a perfect square trinomial.
4. Factor the perfect square trinomial: Rewrite the left side as (x + b/2a)².
5. Take the square root: Take the square root of both sides of the equation.
6. Solve for x: Solve the resulting linear equations to find the values of x.

Example 1:

Solve the quadratic equation: x² + 6x - 7 = 0

1. Rearrange the equation: x² + 6x = 7
2. Divide by a: Since a = 1, no division is needed.
3. Complete the square:
   • (b/2a)² = (6/2)² = 3² = 9
   • x² + 6x + 9 = 7 + 9
4. Factor the perfect square trinomial: (x + 3)² = 16
5. Take the square root: x + 3 = ±√16 x + 3 = ±4
6. Solve for x:
   • x = -3 + 4 = 1
   • x = -3 - 4 = -7

Thus, the solutions are x = 1 and x = -7.

Example 2:

Solve the quadratic equation: 2x² - 8x + 6 = 0

1. Rearrange the equation: 2x² - 8x = -6
2. Divide by a: x² - 4x = -3
3. Complete the square:
   • (b/2a)² = (-4/2)² = (-2)² = 4
   • x² - 4x + 4 = -3 + 4
4. Factor the perfect square trinomial: (x - 2)² = 1
5. Take the square root: x - 2 = ±√1 x - 2 = ±1
6. Solve for x:
   • x = 2 + 1 = 3
   • x = 2 - 1 = 1

Thus, the solutions are x = 3 and x = 1.

3. Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation, regardless of whether it can be easily factored or completed into a perfect square. It provides a direct way to find the solutions using the coefficients a, b, and c from the standard form of the quadratic equation.

The Quadratic Formula:

For a quadratic equation ax² + bx + c = 0, the solutions are given by:

x = (-b ± √(b² - 4ac)) / 2a

Steps for Solving Using the Quadratic Formula:

1. Identify a, b, and c: Determine the values of a, b, and c from the quadratic equation.
2. Substitute into the formula: Plug the values of a, b, and c into the quadratic formula.
3. Simplify: Simplify the expression to find the values of x.

Example 1:

Solve the quadratic equation: x² - 5x + 6 = 0

1. Identify a, b, and c:
   • a = 1
   • b = -5
   • c = 6
2. Substitute into the formula: x = (-(-5) ± √((-5)² - 4(1)(6))) / 2(1)
3. Simplify: x = (5 ± √(25 - 24)) / 2 x = (5 ± √1) / 2 x = (5 ± 1) / 2
   • x = (5 + 1) / 2 = 6 / 2 = 3
   • x = (5 - 1) / 2 = 4 / 2 = 2

Thus, the solutions are x = 3 and x = 2.

Example 2:

Solve the quadratic equation: 2x² + 4x + 1 = 0

1. Identify a, b, and c:
   • a = 2
   • b = 4
   • c = 1
2. Substitute into the formula: x = (-4 ± √(4² - 4(2)(1))) / 2(2)
3. Simplify: x = (-4 ± √(16 - 8)) / 4 x = (-4 ± √8) / 4 x = (-4 ± 2√2) / 4 x = (-2 ± √2) / 2
   • x = (-2 + √2) / 2
   • x = (-2 - √2) / 2

Thus, the solutions are x = (-2 + √2) / 2 and x = (-2 - √2) / 2.

The Discriminant

The discriminant is the part of the quadratic formula under the square root, b² - 4ac. It provides valuable information about the nature of the solutions:

• If b² - 4ac > 0: The equation has two distinct real solutions.
• If b² - 4ac = 0: The equation has one real solution (a repeated root).
• If b² - 4ac < 0: The equation has two complex solutions.

Solving Quadratic Equations: Answer Key Approach

When learning to solve quadratic equations, having an "answer key" approach can be extremely helpful. This involves working through problems with known solutions, allowing you to check your work and understand the steps involved. Here are several quadratic equations with their solutions, serving as an answer key for practice:

Practice Problems and Solutions:

1. Equation: x² - 4x + 3 = 0

   • Solutions: x = 1, x = 3
   • Method: Factoring: (x - 1)(x - 3) = 0

2. Equation: x² + 8x + 16 = 0

   • Solution: x = -4 (repeated root)
   • Method: Factoring: (x + 4)(x + 4) = 0

3. Equation: 2x² - 5x - 3 = 0

   • Solutions: x = 3, x = -1/2
   • Method: Factoring: (2x + 1)(x - 3) = 0

4. Equation: x² + 2x - 1 = 0

   • Solutions: x = -1 + √2, x = -1 - √2
   • Method: Quadratic Formula or Completing the Square

5. Equation: 3x² + 6x + 2 = 0

   • Solutions: x = (-3 + √3) / 3, x = (-3 - √3) / 3
   • Method: Quadratic Formula or Completing the Square

6. Equation: x² - 2x + 5 = 0

   • Solutions: x = 1 + 2i, x = 1 - 2i
   • Method: Quadratic Formula

7. Equation: 4x² - 4x + 1 = 0

   • Solution: x = 1/2 (repeated root)
   • Method: Factoring or Quadratic Formula

8. Equation: x² + 5x + 4 = 0

   • Solutions: x = -1, x = -4
   • Method: Factoring

9. Equation: x² - 9 = 0

   • Solutions: x = 3, x = -3
   • Method: Factoring or Square Root Property

10. Equation: 2x² + 3x - 2 = 0

    • Solutions: x = 1/2, x = -2
    • Method: Factoring

Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications:

1. Physics: Projectile motion, where the height of an object is described by a quadratic equation.
2. Engineering: Designing structures and calculating stress and strain.
3. Economics: Modeling cost, revenue, and profit functions.
4. Computer Science: Algorithm design and optimization problems.
5. Finance: Calculating compound interest and investment returns.

Tips for Mastering Quadratic Equations

1. Practice Regularly: The more you practice, the more comfortable you will become with the different methods.
2. Understand the Concepts: Don't just memorize formulas; understand the underlying principles.
3. Check Your Answers: Use the solutions provided in the answer key or substitute your answers back into the original equation to verify.
4. Use Online Resources: There are many websites and videos available that can provide additional explanations and examples.
5. Seek Help When Needed: Don't hesitate to ask your teacher, tutor, or classmates for help if you are struggling.

Conclusion

Solving quadratic equations is a crucial skill in algebra with widespread applications. By mastering the techniques of factoring, completing the square, and using the quadratic formula, you can tackle a wide range of problems. The "answer key" approach, with practice problems and solutions, provides a valuable tool for self-assessment and reinforcement. Remember to understand the underlying concepts, practice regularly, and seek help when needed. With dedication and perseverance, you can achieve mastery in solving quadratic equations and unlock their potential in various fields.