If Using The Method Of Completing The Square To Solve The Quadratic Equation X 2 + 20 X + 28 = 0 X^2 + 20x + 28 = 0 X 2 + 20 X + 28 = 0 , Which Number Would Have To Be Added To complete The Square?

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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. One of the most effective methods for solving quadratic equations is completing the square. This method involves manipulating the equation to express it in a perfect square trinomial form, which can be easily solved. In this article, we will explore the process of completing the square and determine which number needs to be added to solve the quadratic equation x2+20x+28=0x^2 + 20x + 28 = 0.

What is Completing the Square?

Completing the square is a mathematical technique used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. The method involves manipulating the equation to express it in the form (x+p)2=q(x + p)^2 = q, where pp and qq are constants. This form is called a perfect square trinomial, and it can be easily solved by taking the square root of both sides.

The Process of Completing the Square

To complete the square, we need to follow these steps:

  1. Write the equation in the standard form: The equation should be in the form ax2+bx+c=0ax^2 + bx + c = 0.
  2. Move the constant term to the right-hand side: We need to move the constant term to the right-hand side of the equation to isolate the terms involving xx.
  3. Divide the coefficient of the xx term by 2: We need to divide the coefficient of the xx term by 2 and square the result.
  4. Add the squared result to both sides: We need to add the squared result to both sides of the equation to complete the square.
  5. Simplify the equation: We need to simplify the equation to express it in the form (x+p)2=q(x + p)^2 = q.

Solving the Quadratic Equation x2+20x+28=0x^2 + 20x + 28 = 0

Now, let's apply the process of completing the square to solve the quadratic equation x2+20x+28=0x^2 + 20x + 28 = 0.

Step 1: Write the equation in the standard form

The equation is already in the standard form: x2+20x+28=0x^2 + 20x + 28 = 0.

Step 2: Move the constant term to the right-hand side

We need to move the constant term to the right-hand side of the equation:

x2+20x=−28x^2 + 20x = -28

Step 3: Divide the coefficient of the xx term by 2

We need to divide the coefficient of the xx term by 2 and square the result:

202=10\frac{20}{2} = 10

102=10010^2 = 100

Step 4: Add the squared result to both sides

We need to add the squared result to both sides of the equation to complete the square:

x2+20x+100=−28+100x^2 + 20x + 100 = -28 + 100

x2+20x+100=72x^2 + 20x + 100 = 72

Step 5: Simplify the equation

We need to simplify the equation to express it in the form (x+p)2=q(x + p)^2 = q:

(x+10)2=72(x + 10)^2 = 72

Which Number Needs to be Added to Complete the Square?

To complete the square, we need to add the squared result to both sides of the equation. In this case, the squared result is 100. Therefore, the number that needs to be added to complete the square is 100.

Conclusion

Completing the square is a powerful method for solving quadratic equations. By following the steps outlined in this article, we can easily solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. In this article, we applied the process of completing the square to solve the quadratic equation x2+20x+28=0x^2 + 20x + 28 = 0 and determined that the number that needs to be added to complete the square is 100.

Frequently Asked Questions

Q: What is completing the square?

A: Completing the square is a mathematical technique used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0.

Q: How do I complete the square?

A: To complete the square, you need to follow these steps:

  1. Write the equation in the standard form.
  2. Move the constant term to the right-hand side.
  3. Divide the coefficient of the xx term by 2 and square the result.
  4. Add the squared result to both sides.
  5. Simplify the equation.

Q: What number needs to be added to complete the square for the equation x2+20x+28=0x^2 + 20x + 28 = 0?

A: The number that needs to be added to complete the square is 100.

References

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 completing the square. This method involves manipulating the equation to express it in a perfect square trinomial form, which can be easily solved. In this article, we will explore the process of completing the square and provide a comprehensive guide to solving quadratic equations.

Frequently Asked Questions

Q: What is completing the square?

A: Completing the square is a mathematical technique used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It involves manipulating the equation to express it in the form (x+p)2=q(x + p)^2 = q, where pp and qq are constants.

Q: How do I complete the square?

A: To complete the square, you need to follow these steps:

  1. Write the equation in the standard form.
  2. Move the constant term to the right-hand side.
  3. Divide the coefficient of the xx term by 2 and square the result.
  4. Add the squared result to both sides.
  5. Simplify the equation.

Q: What is the purpose of completing the square?

A: The purpose of completing the square is to express the quadratic equation in a perfect square trinomial form, which can be easily solved.

Q: How do I determine the number to add to complete the square?

A: To determine the number to add to complete the square, you need to divide the coefficient of the xx term by 2 and square the result.

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

A: Completing the square and factoring are two different methods for solving quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while completing the square involves manipulating the equation to express it in a perfect square trinomial form.

Q: Can I use completing the square to solve all types of quadratic equations?

A: No, completing the square is not suitable for all types of quadratic equations. It is particularly useful for quadratic equations that cannot be factored easily.

Q: How do I check my work when completing the square?

A: To check your work, you need to simplify the equation and ensure that it is in the form (x+p)2=q(x + p)^2 = q. You can also use the quadratic formula to check your work.

Common Mistakes to Avoid

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

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

  • Not writing the equation in the standard form.
  • Not moving the constant term to the right-hand side.
  • Not dividing the coefficient of the xx term by 2 and squaring the result.
  • Not adding the squared result to both sides.
  • Not simplifying the equation.

Tips and Tricks

Q: What are some tips and tricks for completing the square?

A: Some tips and tricks for completing the square include:

  • Use a calculator to check your work.
  • Use a graphing calculator to visualize the quadratic equation.
  • Use the quadratic formula to check your work.
  • Practice, practice, practice!

Conclusion

Completing the square is a powerful method for solving quadratic equations. By following the steps outlined in this article, you can easily solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. Remember to avoid common mistakes and use tips and tricks to make the process easier.

References

Additional Resources