Solve $x^2 - 16x + 60 = -12$ By Completing The Steps.First, Subtract $\square$ From Each Side Of The Equation.

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will guide you through the process of solving a quadratic equation by completing the square. We will use the equation $x^2 - 16x + 60 = -12$ as an example.

Step 1: Subtract $\square$ from Each Side of the Equation

The first step in solving a quadratic equation by completing the square is to subtract $\square$ from each side of the equation. This will help us isolate the quadratic term on one side of the equation.

x2−16x+60=−12x^2 - 16x + 60 = -12

Subtracting $\square$ from each side of the equation, we get:

x2−16x+60+□=−12+□x^2 - 16x + 60 + \square = -12 + \square

Simplifying the equation, we get:

x2−16x+60=−12+□x^2 - 16x + 60 = -12 + \square

Step 2: Move the Constant Term to the Right Side of the Equation

The next step is to move the constant term to the right side of the equation. This will help us isolate the quadratic term on one side of the equation.

x2−16x+60=−12+□x^2 - 16x + 60 = -12 + \square

Moving the constant term to the right side of the equation, we get:

x2−16x=−12−60+□x^2 - 16x = -12 - 60 + \square

Simplifying the equation, we get:

x2−16x=−72+□x^2 - 16x = -72 + \square

Step 3: Add and Subtract the Square of Half the Coefficient of the Linear Term

The next step is to add and subtract the square of half the coefficient of the linear term. This will help us create a perfect square trinomial on the left side of the equation.

x2−16x=−72+□x^2 - 16x = -72 + \square

The coefficient of the linear term is -16. Half of -16 is -8. The square of -8 is 64.

Adding and subtracting 64 to the left side of the equation, we get:

x2−16x+64=−72+64+□x^2 - 16x + 64 = -72 + 64 + \square

Simplifying the equation, we get:

x2−16x+64=−8+□x^2 - 16x + 64 = -8 + \square

Step 4: Factor the Perfect Square Trinomial

The next step is to factor the perfect square trinomial on the left side of the equation.

x2−16x+64=−8+□x^2 - 16x + 64 = -8 + \square

The perfect square trinomial can be factored as:

(x−8)2=−8+□(x - 8)^2 = -8 + \square

Step 5: Simplify the Right Side of the Equation

The next step is to simplify the right side of the equation.

(x−8)2=−8+□(x - 8)^2 = -8 + \square

Simplifying the right side of the equation, we get:

(x−8)2=64(x - 8)^2 = 64

Step 6: Take the Square Root of Both Sides of the Equation

The next step is to take the square root of both sides of the equation.

(x−8)2=64(x - 8)^2 = 64

Taking the square root of both sides of the equation, we get:

x−8=±64x - 8 = \pm \sqrt{64}

Simplifying the equation, we get:

x−8=±8x - 8 = \pm 8

Step 7: Add 8 to Both Sides of the Equation

The final step is to add 8 to both sides of the equation.

x−8=±8x - 8 = \pm 8

Adding 8 to both sides of the equation, we get:

x=8±8x = 8 \pm 8

Simplifying the equation, we get:

x = 16$ or $x = 0

Conclusion

In this article, we have guided you through the process of solving a quadratic equation by completing the square. We have used the equation $x^2 - 16x + 60 = -12$ as an example. By following the steps outlined in this article, you should be able to solve quadratic equations by completing the square.

Common Mistakes to Avoid

When solving quadratic equations by completing the square, there are several common mistakes to avoid. These include:

  • Not subtracting $\square$ from each side of the equation
  • Not moving the constant term to the right side of the equation
  • Not adding and subtracting the square of half the coefficient of the linear term
  • Not factoring the perfect square trinomial
  • Not simplifying the right side of the equation
  • Not taking the square root of both sides of the equation
  • Not adding 8 to both sides of the equation

By avoiding these common mistakes, you should be able to solve quadratic equations by completing the square with ease.

Real-World Applications

Quadratic equations have numerous real-world applications. These include:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design bridges and buildings.
  • Economics: Quadratic equations are used to model the behavior of economic systems.
  • Computer Science: Quadratic equations are used to solve problems in computer science, such as finding the shortest path between two nodes in a graph.

By understanding how to solve quadratic equations by completing the square, you will be able to apply this knowledge to a wide range of real-world problems.

Final Thoughts

Introduction

In our previous article, we guided you through the process of solving quadratic equations by completing the square. In this article, we will answer some of the most frequently asked questions about quadratic equations by completing the square.

Q: What is completing the square?

A: Completing the square is a method of solving quadratic equations by rewriting them in a perfect square trinomial form. This allows us to easily factor the equation and solve for the variable.

Q: Why do we need to complete the square?

A: We need to complete the square because it allows us to easily factor the equation and solve for the variable. This is especially useful when the equation is not easily factorable.

Q: What are the steps to complete the square?

A: The steps to complete the square are:

  1. Subtract $\square$ from each side of the equation.
  2. Move the constant term to the right side of the equation.
  3. Add and subtract the square of half the coefficient of the linear term.
  4. Factor the perfect square trinomial.
  5. Simplify the right side of the equation.
  6. Take the square root of both sides of the equation.
  7. Add 8 to both sides of the equation.

Q: What is the difference between completing the square and factoring?

A: Completing the square and factoring are two different methods of solving quadratic equations. Factoring involves finding two binomials whose product is the original equation, while completing the square involves rewriting the equation in a perfect square trinomial form.

Q: When should I use completing the square?

A: You should use completing the square when the equation is not easily factorable, or when you need to find the solutions to the equation.

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

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

  • Not subtracting $\square$ from each side of the equation
  • Not moving the constant term to the right side of the equation
  • Not adding and subtracting the square of half the coefficient of the linear term
  • Not factoring the perfect square trinomial
  • Not simplifying the right side of the equation
  • Not taking the square root of both sides of the equation
  • Not adding 8 to both sides of the equation

Q: How do I know if I should use completing the square or factoring?

A: You should use completing the square if the equation is not easily factorable, or if you need to find the solutions to the equation. You should use factoring if the equation is easily factorable.

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

A: Yes, you can use completing the square to solve quadratic equations with complex coefficients. However, you will need to use complex numbers and follow the same steps as before.

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

A: Yes, you can use completing the square to solve quadratic equations with rational coefficients. However, you will need to follow the same steps as before and simplify the equation.

Conclusion

In this article, we have answered some of the most frequently asked questions about quadratic equations by completing the square. We hope that this guide has been helpful in understanding the process of completing the square and how to apply it to solve quadratic equations.

Real-World Applications

Quadratic equations have numerous real-world applications. These include:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design bridges and buildings.
  • Economics: Quadratic equations are used to model the behavior of economic systems.
  • Computer Science: Quadratic equations are used to solve problems in computer science, such as finding the shortest path between two nodes in a graph.

By understanding how to solve quadratic equations by completing the square, you will be able to apply this knowledge to a wide range of real-world problems.

Final Thoughts

In conclusion, solving quadratic equations by completing the square is a powerful tool that can be used to solve a wide range of problems. By following the steps outlined in this article, you should be able to solve quadratic equations by completing the square with ease. Remember to avoid common mistakes and apply this knowledge to real-world problems. With practice and patience, you will become proficient in solving quadratic equations by completing the square.