Edward Used Completing The Square To Find The Minimum Value Of The Expression $x^2+2x+5$. Which Is The Equivalent Expression After Completing The Square?A. $(x+1)^2+5$B. $ ( X + 1 ) 2 + 4 (x+1)^2+4 ( X + 1 ) 2 + 4 [/tex]C. $(x+2)^2+1$D.

Introduction

Completing the square is a fundamental technique in algebra that allows us to rewrite a quadratic expression in a form that reveals its minimum or maximum value. This technique is particularly useful in solving quadratic equations and inequalities. In this article, we will explore the concept of completing the square and apply it to find the minimum value of the expression $x^2+2x+5$.

What is Completing the Square?

Completing the square is a method of rewriting a quadratic expression in the form $(x+a)^2+b$. This form is useful because it reveals the minimum or maximum value of the expression. The process of completing the square involves manipulating the quadratic expression to create a perfect square trinomial.

The Process of Completing the Square

To complete the square, we follow these steps:

1. Write the quadratic expression in the form $x^2+bx+c$. In this case, the expression is $x^2+2x+5$.
2. Take the coefficient of the $x$ term, which is $b$, and divide it by 2. In this case, $b=2$, so we divide it by 2 to get $1$.
3. Square the result from step 2. In this case, we square $1$ to get $1$.
4. Add the result from step 3 to both sides of the expression. In this case, we add $1$ to both sides of the expression $x^2+2x+5$.
5. Write the result from step 4 in the form $(x+a)^2+b$. In this case, we write the result as $(x+1)^2+5$.

Applying Completing the Square to the Expression $x^2+2x+5$

Now that we have understood the process of completing the square, let's apply it to the expression $x^2+2x+5$. We follow the steps outlined above:

1. Write the quadratic expression in the form $x^2+bx+c$. In this case, the expression is $x^2+2x+5$.
2. Take the coefficient of the $x$ term, which is $b$, and divide it by 2. In this case, $b=2$, so we divide it by 2 to get $1$.
3. Square the result from step 2. In this case, we square $1$ to get $1$.
4. Add the result from step 3 to both sides of the expression. In this case, we add $1$ to both sides of the expression $x^2+2x+5$.
5. Write the result from step 4 in the form $(x+a)^2+b$. In this case, we write the result as $(x+1)^2+5$.

Conclusion

In this article, we have explored the concept of completing the square and applied it to find the minimum value of the expression $x^2+2x+5$. We have seen that completing the square involves manipulating the quadratic expression to create a perfect square trinomial. By following the steps outlined above, we can rewrite the expression in the form $(x+a)^2+b$, which reveals its minimum or maximum value.

Answer

The equivalent expression after completing the square is:

$$(x+1)^2+5$$

This is option A.

Discussion

Completing the square is a powerful technique in algebra that allows us to rewrite a quadratic expression in a form that reveals its minimum or maximum value. This technique is particularly useful in solving quadratic equations and inequalities. In this article, we have seen how to apply completing the square to find the minimum value of the expression $x^2+2x+5$. We have also seen how to write the result in the form $(x+a)^2+b$, which reveals its minimum or maximum value.

References

• Completing the Square
• Quadratic Equations
• Algebra

Related Topics

• Solving Quadratic Equations
• Graphing Quadratic Functions
• Systems of Linear Equations

Frequently Asked Questions

• What is completing the square?
  • Completing the square is a method of rewriting a quadratic expression in the form $(x+a)^2+b$.
• How do I apply completing the square to a quadratic expression?
  • To apply completing the square, follow the steps outlined above: write the quadratic expression in the form $x^2+bx+c$, take the coefficient of the $x$ term and divide it by 2, square the result, add the result to both sides of the expression, and write the result in the form $(x+a)^2+b$.
• What is the minimum value of the expression $x^2+2x+5$?
  • The minimum value of the expression $x^2+2x+5$ is 4.
    Completing the Square: A Powerful Technique in Algebra ===========================================================

Q&A: Completing the Square

Q: What is completing the square?

A: Completing the square is a method of rewriting a quadratic expression in the form $(x+a)^2+b$. This form is useful because it reveals the minimum or maximum value of the expression.

Q: How do I apply completing the square to a quadratic expression?

A: To apply completing the square, follow these steps:

1. Write the quadratic expression in the form $x^2+bx+c$. In this case, the expression is $x^2+2x+5$.
2. Take the coefficient of the $x$ term, which is $b$, and divide it by 2. In this case, $b=2$, so we divide it by 2 to get $1$.
3. Square the result from step 2. In this case, we square $1$ to get $1$.
4. Add the result from step 3 to both sides of the expression. In this case, we add $1$ to both sides of the expression $x^2+2x+5$.
5. Write the result from step 4 in the form $(x+a)^2+b$. In this case, we write the result as $(x+1)^2+5$.

Q: What is the minimum value of the expression $x^2+2x+5$?

A: The minimum value of the expression $x^2+2x+5$ is 4.

Q: How do I find the minimum value of a quadratic expression using completing the square?

A: To find the minimum value of a quadratic expression using completing the square, follow these steps:

1. Write the quadratic expression in the form $x^2+bx+c$. In this case, the expression is $x^2+2x+5$.
2. Take the coefficient of the $x$ term, which is $b$, and divide it by 2. In this case, $b=2$, so we divide it by 2 to get $1$.
3. Square the result from step 2. In this case, we square $1$ to get $1$.
4. Add the result from step 3 to both sides of the expression. In this case, we add $1$ to both sides of the expression $x^2+2x+5$.
5. Write the result from step 4 in the form $(x+a)^2+b$. In this case, we write the result as $(x+1)^2+5$.
6. The minimum value of the expression is the value of $b$ in the form $(x+a)^2+b$. In this case, the minimum value is 5.

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

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

• Not following the steps correctly. Make sure to follow the steps outlined above to complete the square correctly.
• Not squaring the result correctly. Make sure to square the result from step 2 correctly.
• Not adding the result to both sides of the expression correctly. Make sure to add the result to both sides of the expression correctly.
• Not writing the result in the correct form. Make sure to write the result in the form $(x+a)^2+b$.

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

A: To check your work when completing the square, follow these steps:

1. Write the original expression and the completed square expression. In this case, the original expression is $x^2+2x+5$ and the completed square expression is $(x+1)^2+5$.
2. Check that the completed square expression is in the correct form. In this case, the completed square expression is in the correct form $(x+a)^2+b$.
3. Check that the minimum value of the expression is correct. In this case, the minimum value of the expression is 4.

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

A: Some real-world applications of completing the square include:

• Optimization problems. Completing the square can be used to solve optimization problems, such as finding the maximum or minimum value of a function.
• Physics and engineering problems. Completing the square can be used to solve physics and engineering problems, such as finding the trajectory of a projectile or the motion of an object.
• Computer science problems. Completing the square can be used to solve computer science problems, such as finding the shortest path between two points or the minimum cost of a network.

Conclusion

In this article, we have explored the concept of completing the square and answered some common questions about this technique. We have seen how to apply completing the square to find the minimum value of a quadratic expression and how to check our work when completing the square. We have also seen some real-world applications of completing the square.