Rewrite The Function By Completing The Square.$\[ \begin{array}{l} g(x) = X^2 + 15x + 54 \\ g(x) = \square (x + \square)^2 + \square \end{array} \\]

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Introduction

Completing the square is a powerful technique used in algebra to rewrite quadratic expressions in a more convenient form. It involves manipulating the expression to create a perfect square trinomial, which can be factored into the square of a binomial. In this article, we will demonstrate how to rewrite the function g(x) = x^2 + 15x + 54 by completing the square.

The Process of Completing the Square

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

  1. Start with the given quadratic expression: In this case, we have g(x) = x^2 + 15x + 54.
  2. Take the coefficient of the x-term and divide it by 2: The coefficient of the x-term is 15, so we divide it by 2 to get 7.5.
  3. Square the result: We square 7.5 to get 56.25.
  4. Add and subtract the squared result: We add and subtract 56.25 inside the expression to create a perfect square trinomial.

Rewriting the Function by Completing the Square

Now, let's apply the steps to rewrite the function g(x) = x^2 + 15x + 54 by completing the square.

Step 1: Start with the given quadratic expression

g(x) = x^2 + 15x + 54

Step 2: Take the coefficient of the x-term and divide it by 2

The coefficient of the x-term is 15, so we divide it by 2 to get 7.5.

Step 3: Square the result

We square 7.5 to get 56.25.

Step 4: Add and subtract the squared result

We add and subtract 56.25 inside the expression to create a perfect square trinomial.

g(x) = (x^2 + 15x + 56.25) - 56.25 + 54

Step 5: Factor the perfect square trinomial

We can factor the perfect square trinomial as follows:

g(x) = (x + 7.5)^2 - 2.25 + 54

Step 6: Simplify the expression

We can simplify the expression by combining the constants:

g(x) = (x + 7.5)^2 + 51.75

Step 7: Write the final expression

The final expression is:

g(x) = (x + 7.5)^2 + 51.75

Conclusion

In this article, we demonstrated how to rewrite the function g(x) = x^2 + 15x + 54 by completing the square. We followed the steps of taking the coefficient of the x-term, squaring the result, adding and subtracting the squared result, factoring the perfect square trinomial, and simplifying the expression. The final expression is g(x) = (x + 7.5)^2 + 51.75.

Example Use Cases

Completing the square has many applications in mathematics and other fields. Here are a few example use cases:

  • Graphing quadratic functions: Completing the square can help us graph quadratic functions by rewriting them in vertex form.
  • Solving quadratic equations: Completing the square can help us solve quadratic equations by rewriting them in a form that is easier to solve.
  • Optimization problems: Completing the square can help us solve optimization problems by rewriting the objective function in a form that is easier to minimize or maximize.

Tips and Tricks

Here are a few tips and tricks to help you complete the square:

  • Make sure to square the result: When completing the square, make sure to square the result of the coefficient of the x-term divided by 2.
  • Add and subtract the squared result: When adding and subtracting the squared result, make sure to add and subtract the same value.
  • Factor the perfect square trinomial: When factoring the perfect square trinomial, make sure to factor it as the square of a binomial.

Common Mistakes

Here are a few common mistakes to avoid when completing the square:

  • Not squaring the result: Failing to square the result of the coefficient of the x-term divided by 2 can lead to incorrect results.
  • Not adding and subtracting the squared result: Failing to add and subtract the squared result can lead to incorrect results.
  • Not factoring the perfect square trinomial: Failing to factor the perfect square trinomial can lead to incorrect results.

Conclusion

Introduction

Completing the square is a powerful technique used in algebra to rewrite quadratic expressions in a more convenient form. In our previous article, we demonstrated how to rewrite the function g(x) = x^2 + 15x + 54 by completing the square. 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 technique used to rewrite a quadratic expression in a form that is easier to work with. It involves manipulating the expression to create a perfect square trinomial, which can be factored into the square of a binomial.

Q: Why is completing the square useful?

A: Completing the square is useful because it allows us to rewrite quadratic expressions in a form that is easier to work with. This can be helpful when graphing quadratic functions, solving quadratic equations, and optimizing problems.

Q: How do I complete the square?

A: To complete the square, follow these steps:

  1. Start with the given quadratic expression: In this case, we have g(x) = x^2 + 15x + 54.
  2. Take the coefficient of the x-term and divide it by 2: The coefficient of the x-term is 15, so we divide it by 2 to get 7.5.
  3. Square the result: We square 7.5 to get 56.25.
  4. Add and subtract the squared result: We add and subtract 56.25 inside the expression to create a perfect square trinomial.

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a trinomial that can be factored into the square of a binomial. It has the form (x + a)^2, where a is a constant.

Q: How do I factor a perfect square trinomial?

A: To factor a perfect square trinomial, follow these steps:

  1. Identify the binomial: The binomial is the expression inside the parentheses.
  2. Square the binomial: We square the binomial to get the perfect square trinomial.
  3. Write the final expression: The final expression is the perfect square trinomial.

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

A: Here are a few common mistakes to avoid when completing the square:

  • Not squaring the result: Failing to square the result of the coefficient of the x-term divided by 2 can lead to incorrect results.
  • Not adding and subtracting the squared result: Failing to add and subtract the squared result can lead to incorrect results.
  • Not factoring the perfect square trinomial: Failing to factor the perfect square trinomial can lead to incorrect results.

Q: How do I know if I have completed the square correctly?

A: To check if you have completed the square correctly, follow these steps:

  1. Check the expression: Make sure the expression is in the form (x + a)^2 + b.
  2. Check the binomial: Make sure the binomial is correct.
  3. Check the constant: Make sure the constant is correct.

Q: What are some real-world applications of completing the square?

A: Completing the square has many real-world applications, including:

  • Graphing quadratic functions: Completing the square can help us graph quadratic functions by rewriting them in vertex form.
  • Solving quadratic equations: Completing the square can help us solve quadratic equations by rewriting them in a form that is easier to solve.
  • Optimization problems: Completing the square can help us solve optimization problems by rewriting the objective function in a form that is easier to minimize or maximize.

Conclusion

In conclusion, completing the square is a powerful technique used in algebra to rewrite quadratic expressions in a more convenient form. By following the steps of taking the coefficient of the x-term, squaring the result, adding and subtracting the squared result, factoring the perfect square trinomial, and simplifying the expression, we can rewrite the function g(x) = x^2 + 15x + 54 by completing the square. We hope this Q&A guide has been helpful in answering your questions about completing the square.