What Number Should Be Added To Both Sides Of The Equation To Complete The Square?${ X^2 + 8x = 4 }$A. 4 B. 8 C. 16 D. 32

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What Number Should Be Added to Both Sides of the Equation to Complete the Square?

Understanding the Concept of Completing the Square

Completing the square is a mathematical technique used to rewrite a quadratic equation in a specific form, which makes it easier to solve. This method involves adding a constant term to both sides of the equation to create a perfect square trinomial. The constant term added is determined by the coefficient of the linear term in the quadratic equation.

The Formula for Completing the Square

The formula for completing the square is based on the concept that a perfect square trinomial can be written in the form of (x+a)2{(x + a)^2}, where a{a} is a constant. To complete the square, we need to add and subtract (b2)2{(\frac{b}{2})^2} to the equation, where b{b} is the coefficient of the linear term.

Applying the Formula to the Given Equation

In the given equation, x2+8x=4{x^2 + 8x = 4}, the coefficient of the linear term is 8. To complete the square, we need to add and subtract (82)2=16{(\frac{8}{2})^2 = 16} to the equation.

Step 1: Add 16 to Both Sides of the Equation

x2+8x+16=4+16{ x^2 + 8x + 16 = 4 + 16 }

Step 2: Simplify the Equation

(x+4)2=20{ (x + 4)^2 = 20 }

Step 3: Solve for x

x+4=±20{ x + 4 = \pm \sqrt{20} }

x=−4±20{ x = -4 \pm \sqrt{20} }

Determining the Correct Answer

To determine the correct answer, we need to analyze the options provided. The options are 4, 8, 16, and 32. Based on the steps we followed to complete the square, we added 16 to both sides of the equation. Therefore, the correct answer is 16.

Conclusion

In conclusion, to complete the square, we need to add and subtract a constant term to the equation. The constant term added is determined by the coefficient of the linear term in the quadratic equation. In this case, we added 16 to both sides of the equation to complete the square. Therefore, the correct answer is 16.

Frequently Asked Questions

  • What is completing the square? Completing the square is a mathematical technique used to rewrite a quadratic equation in a specific form, which makes it easier to solve.
  • How do I complete the square? To complete the square, you need to add and subtract a constant term to the equation, which is determined by the coefficient of the linear term.
  • What is the formula for completing the square? The formula for completing the square is based on the concept that a perfect square trinomial can be written in the form of (x+a)2{(x + a)^2}, where a{a} is a constant.

Additional Resources

  • Khan Academy: Completing the Square
  • Mathway: Completing the Square
  • Wolfram Alpha: Completing the Square

References

  • "Algebra and Trigonometry" by Michael Sullivan
  • "College Algebra" by James Stewart
  • "Precalculus" by Michael Sullivan
    Completing the Square: A Q&A Article

Understanding Completing the Square

Completing the square is a mathematical technique used to rewrite a quadratic equation in a specific form, which makes it easier to solve. This method involves adding a constant term to both sides of the equation to create a perfect square trinomial. In this article, we will answer some frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a mathematical technique used to rewrite a quadratic equation in a specific form, which makes it easier to solve. This method involves adding a constant term to both sides of the equation to create a perfect square trinomial.

Q: How do I complete the square?

A: To complete the square, you need to add and subtract a constant term to the equation, which is determined by the coefficient of the linear term. The formula for completing the square is based on the concept that a perfect square trinomial can be written in the form of (x+a)2{(x + a)^2}, where a{a} is a constant.

Q: What is the formula for completing the square?

A: The formula for completing the square is based on the concept that a perfect square trinomial can be written in the form of (x+a)2{(x + a)^2}, where a{a} is a constant. To complete the square, you need to add and subtract (b2)2{(\frac{b}{2})^2} to the equation, where b{b} is the coefficient of the linear term.

Q: How do I determine the constant term to add?

A: To determine the constant term to add, you need to find the coefficient of the linear term in the quadratic equation. Then, you need to square half of this coefficient and add it to both sides of the equation.

Q: What is the purpose of completing the square?

A: The purpose of completing the square is to rewrite a quadratic equation in a specific form, which makes it easier to solve. This method is useful for solving quadratic equations that cannot be factored easily.

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. This method is only useful for quadratic equations that can be written in the form of ax2+bx+c=0{ax^2 + bx + c = 0}, where a{a}, b{b}, and c{c} are constants.

Q: How do I know if I should use completing the square or factoring to solve a quadratic equation?

A: To determine whether to use completing the square or factoring, you need to examine the quadratic equation and see if it can be factored easily. If it can be factored easily, then factoring is the better option. If it cannot be factored easily, then completing the square is the better option.

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

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

  • Adding the constant term to only one side of the equation
  • Subtracting the constant term from only one side of the equation
  • Not squaring half of the coefficient of the linear term
  • Not adding and subtracting the constant term to both sides of the equation

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

A: To check your work when completing the square, you need to verify that the equation is in the form of (x+a)2=b{(x + a)^2 = b}, where a{a} and b{b} are constants. You can do this by expanding the left-hand side of the equation and comparing it to the right-hand side.

Conclusion

In conclusion, completing the square is a mathematical technique used to rewrite a quadratic equation in a specific form, which makes it easier to solve. This method involves adding a constant term to both sides of the equation to create a perfect square trinomial. By following the steps outlined in this article, you can learn how to complete the square and solve quadratic equations more easily.

Frequently Asked Questions

  • What is completing the square?
  • How do I complete the square?
  • What is the formula for completing the square?
  • How do I determine the constant term to add?
  • What is the purpose of completing the square?
  • Can I use completing the square to solve all types of quadratic equations?
  • How do I know if I should use completing the square or factoring to solve a quadratic equation?
  • What are some common mistakes to avoid when completing the square?
  • How do I check my work when completing the square?

Additional Resources

  • Khan Academy: Completing the Square
  • Mathway: Completing the Square
  • Wolfram Alpha: Completing the Square

References

  • "Algebra and Trigonometry" by Michael Sullivan
  • "College Algebra" by James Stewart
  • "Precalculus" by Michael Sullivan