Examine How The Completing-the-square Method Is Used To Solve The Quadratic Equation X 2 − 4 X − 9 = 0 X^2 - 4x - 9 = 0 X 2 − 4 X − 9 = 0 Below.Step 1: X 2 − 4 X − 9 + 13 = 0 + 13 X^2 - 4x - 9 + 13 = 0 + 13 X 2 − 4 X − 9 + 13 = 0 + 13 Step 2: X 2 − 4 X + 4 = 13 X^2 - 4x + 4 = 13 X 2 − 4 X + 4 = 13 Step 3: ( X − 2 ) 2 = 13 (x - 2)^2 = 13 ( X − 2 ) 2 = 13 Step 4:

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. One of the most effective methods for solving quadratic equations is the completing-the-square method. In this article, we will examine how the completing-the-square method is used to solve the quadratic equation $x^2 - 4x - 9 = 0$.

Step 1: Adding a Constant to Both Sides

The first step in completing the square is to add a constant to both sides of the equation. This constant is chosen such that it creates a perfect square trinomial on the left-hand side of the equation. In this case, we add 13 to both sides of the equation:

$x^2 - 4x - 9 + 13 = 0 + 13$

This step is crucial because it allows us to create a perfect square trinomial on the left-hand side of the equation.

Step 2: Creating a Perfect Square Trinomial

The next step is to create a perfect square trinomial on the left-hand side of the equation. To do this, we need to add and subtract the square of half the coefficient of the $x$ term. In this case, the coefficient of the $x$ term is -4, so we need to add and subtract $(\frac{-4}{2})^2 = 4$:

$x^2 - 4x + 4 = 13$

By adding and subtracting 4, we have created a perfect square trinomial on the left-hand side of the equation.

Step 3: Factoring the Perfect Square Trinomial

The next step is to factor the perfect square trinomial on the left-hand side of the equation. In this case, we can factor the perfect square trinomial as follows:

$(x - 2)^2 = 13$

By factoring the perfect square trinomial, we have simplified the equation and made it easier to solve.

Step 4: Solving for x

The final step is to solve for x. To do this, we need to take the square root of both sides of the equation. Since we are dealing with a quadratic equation, we need to consider both the positive and negative square roots:

$x - 2 = \pm \sqrt{13}$

By adding 2 to both sides of the equation, we can solve for x:

$x = 2 \pm \sqrt{13}$

Discussion

The completing-the-square method is a powerful tool for solving quadratic equations. By adding a constant to both sides of the equation, creating a perfect square trinomial, factoring the perfect square trinomial, and solving for x, we can solve quadratic equations in a straightforward and efficient manner.

Advantages of the Completing-the-Square Method

The completing-the-square method has several advantages over other methods for solving quadratic equations. Some of the advantages include:

• Efficiency: The completing-the-square method is often faster and more efficient than other methods for solving quadratic equations.
• Accuracy: The completing-the-square method is highly accurate and can be used to solve quadratic equations with high precision.
• Flexibility: The completing-the-square method can be used to solve quadratic equations with any coefficient.

Common Applications of the Completing-the-Square Method

The completing-the-square method has a wide range of applications in mathematics and other fields. Some of the common applications include:

• Solving quadratic equations: The completing-the-square method is used to solve quadratic equations in algebra and other branches of mathematics.
• Graphing quadratic functions: The completing-the-square method is used to graph quadratic functions in mathematics and other fields.
• Optimization problems: The completing-the-square method is used to solve optimization problems in mathematics and other fields.

Conclusion

In conclusion, the completing-the-square method is a powerful tool for solving quadratic equations. By adding a constant to both sides of the equation, creating a perfect square trinomial, factoring the perfect square trinomial, and solving for x, we can solve quadratic equations in a straightforward and efficient manner. The advantages of the completing-the-square method include efficiency, accuracy, and flexibility, making it a popular choice for solving quadratic equations.

Frequently Asked Questions

Q: What is the completing-the-square method?

A: The completing-the-square method is a technique used to solve quadratic equations by adding a constant to both sides of the equation, creating a perfect square trinomial, factoring the perfect square trinomial, and solving for x.

Q: How do I use the completing-the-square method to solve a quadratic equation?

A: To use the completing-the-square method to solve a quadratic equation, follow these steps:

1. Add a constant to both sides of the equation.
2. Create a perfect square trinomial on the left-hand side of the equation.
3. Factor the perfect square trinomial.
4. Solve for x.

Q: What are the advantages of the completing-the-square method?

A: The advantages of the completing-the-square method include efficiency, accuracy, and flexibility.

Q: What are some common applications of the completing-the-square method?

Q: What is the completing-the-square method?

A: The completing-the-square method is a technique used to solve quadratic equations by adding a constant to both sides of the equation, creating a perfect square trinomial, factoring the perfect square trinomial, and solving for x.

Q: How do I use the completing-the-square method to solve a quadratic equation?

A: To use the completing-the-square method to solve a quadratic equation, follow these steps:

1. Add a constant to both sides of the equation.
2. Create a perfect square trinomial on the left-hand side of the equation.
3. Factor the perfect square trinomial.
4. Solve for x.

Q: What are the advantages of the completing-the-square method?

A: The advantages of the completing-the-square method include:

• Efficiency: The completing-the-square method is often faster and more efficient than other methods for solving quadratic equations.
• Accuracy: The completing-the-square method is highly accurate and can be used to solve quadratic equations with high precision.
• Flexibility: The completing-the-square method can be used to solve quadratic equations with any coefficient.

Q: What are some common applications of the completing-the-square method?

A: Some common applications of the completing-the-square method include:

• Solving quadratic equations: The completing-the-square method is used to solve quadratic equations in algebra and other branches of mathematics.
• Graphing quadratic functions: The completing-the-square method is used to graph quadratic functions in mathematics and other fields.
• Optimization problems: The completing-the-square method is used to solve optimization problems in mathematics and other fields.

Q: Can I use the completing-the-square method to solve quadratic equations with complex coefficients?

A: Yes, the completing-the-square method can be used to solve quadratic equations with complex coefficients. However, the process may be more complicated and require additional steps.

Q: How do I choose the correct constant to add to both sides of the equation?

A: To choose the correct constant to add to both sides of the equation, you need to find the square of half the coefficient of the x term. This will ensure that you create a perfect square trinomial on the left-hand side of the equation.

Q: Can I use the completing-the-square method to solve quadratic equations with rational coefficients?

A: Yes, the completing-the-square method can be used to solve quadratic equations with rational coefficients. However, the process may be more complicated and require additional steps.

Q: What are some common mistakes to avoid when using the completing-the-square method?

A: Some common mistakes to avoid when using the completing-the-square method include:

• Not adding the correct constant to both sides of the equation
• Not creating a perfect square trinomial on the left-hand side of the equation
• Not factoring the perfect square trinomial correctly

Q: Can I use the completing-the-square method to solve quadratic equations with negative coefficients?

A: Yes, the completing-the-square method can be used to solve quadratic equations with negative coefficients. However, the process may be more complicated and require additional steps.

Q: How do I know if the completing-the-square method is the best method to use for a particular quadratic equation?

A: To determine if the completing-the-square method is the best method to use for a particular quadratic equation, you need to consider the following factors:

• The complexity of the equation: If the equation is simple and has a small number of terms, the completing-the-square method may be the best choice.
• The type of solution required: If you need to find the exact solution to the equation, the completing-the-square method may be the best choice.
• The time and effort required: If you need to solve the equation quickly and efficiently, the completing-the-square method may be the best choice.

Conclusion

In conclusion, the completing-the-square method is a powerful tool for solving quadratic equations. By understanding the advantages and disadvantages of the method, as well as the common applications and mistakes to avoid, you can use the completing-the-square method to solve quadratic equations with confidence.