Solve by Completing the Square Made Easy

What is Completing the Square?

Completing the square is a powerful technique used in algebra to solve quadratic equations of the form ax^2 + bx + c = 0. This method involves manipulating the equation to express it in a perfect square form, allowing us to easily find the solutions. In this blog post, we will break down the steps to complete the square and provide examples to illustrate the process.

Why is Completing the Square Important?

Completing the square is a fundamental technique in algebra that has numerous applications in various fields, including physics, engineering, and computer science. It is used to solve quadratic equations, which are essential in modeling real-world problems, such as projectile motion, electrical circuits, and population growth. By mastering the art of completing the square, you will be able to tackle a wide range of problems with confidence.

Step-by-Step Guide to Completing the Square

To complete the square, follow these simple steps:

1. Start with a quadratic equation: Begin with a quadratic equation in the form ax^2 + bx + c = 0.
2. Move the constant term: Move the constant term © to the right-hand side of the equation.
3. Divide by the coefficient of x^2: Divide both sides of the equation by the coefficient of x^2 (a).
4. Take half of the coefficient of x: Take half of the coefficient of x (b) and square it.
5. Add and subtract the result: Add and subtract the result from step 4 to the left-hand side of the equation.
6. Factor the perfect square: Factor the perfect square trinomial on the left-hand side of the equation.
7. Solve for x: Solve for x by setting each factor equal to zero.

Example 1: Solving a Quadratic Equation by Completing the Square

Solve the quadratic equation x^2 + 6x + 8 = 0 by completing the square.

1. Move the constant term: x^2 + 6x = -8
2. Divide by the coefficient of x^2: x^2 + 6x = -8 / 1
3. Take half of the coefficient of x: 6 / 2 = 3
4. Add and subtract the result: x^2 + 6x + 3^2 - 3^2 = -8
5. Factor the perfect square: (x + 3)^2 - 9 = -8
6. Solve for x: (x + 3)^2 = 1 x + 3 = ±1 x = -3 ± 1 x = -2 or x = -4

Example 2: Solving a Quadratic Equation with a Coefficient of x^2

Solve the quadratic equation 2x^2 + 4x + 1 = 0 by completing the square.

1. Move the constant term: 2x^2 + 4x = -1
2. Divide by the coefficient of x^2: x^2 + 2x = -1 / 2
3. Take half of the coefficient of x: 2 / 2 = 1
4. Add and subtract the result: x^2 + 2x + 1^2 - 1^2 = -1 / 2
5. Factor the perfect square: (x + 1)^2 - 1 = -1 / 2
6. Solve for x: (x + 1)^2 = 1 / 2 x + 1 = ±√(¹⁄₂) x = -1 ± √(¹⁄₂)

Common Mistakes to Avoid

When completing the square, be careful not to make the following mistakes:

• Forgetting to divide by the coefficient of x^2
• Not taking half of the coefficient of x
• Not adding and subtracting the result correctly
• Not factoring the perfect square trinomial correctly

🚨 Note: Completing the square is a technique that requires practice to master. Make sure to practice with different types of quadratic equations to become proficient.

Conclusion

Completing the square is a powerful technique used to solve quadratic equations. By following the step-by-step guide outlined in this blog post, you will be able to tackle a wide range of problems with confidence. Remember to practice regularly and avoid common mistakes to become proficient in completing the square.

Related Terms:

• Solve by quadratic formula Worksheet