What Number Should Be Added To Both Sides Of The Equation To Complete The Square In X 2 + 8 X = 17 X^2 + 8x = 17 X 2 + 8 X = 17 ?

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Introduction

Completing the square is a powerful technique used in algebra to solve quadratic equations. It involves manipulating the equation to express it in a perfect square form, which can be easily solved. In this article, we will explore how to complete the square in the equation $x^2 + 8x = 17$ and determine the number that should be added to both sides of the equation.

Understanding the Concept of Completing the Square

Completing the square is a method of solving quadratic equations by rewriting them in a perfect square form. This involves adding and subtracting a constant term to the equation, which allows us to express it as a perfect square trinomial. The constant term added is called the "constant of completion" or "constant of perfect square."

The Formula for Completing the Square

The formula for completing the square is:

$$x^2 + bx = (x + \frac{b}{2})^2 - (\frac{b}{2})^2$$

where $b$ is the coefficient of the $x$ term in the quadratic equation.

Applying the Formula to the Given Equation

In the given equation $x^2 + 8x = 17$, we can see that the coefficient of the $x$ term is $8$. Using the formula for completing the square, we can rewrite the equation as:

$$(x + \frac{8}{2})^2 - (\frac{8}{2})^2 = 17$$

Simplifying the equation, we get:

$$(x + 4)^2 - 16 = 17$$

Determining the Number to Be Added to Both Sides

To complete the square, we need to add the constant of completion to both sides of the equation. In this case, the constant of completion is $16$, which is the square of the coefficient of the $x$ term divided by $4$. Therefore, we need to add $16$ to both sides of the equation to complete the square.

The Final Equation

Adding $16$ to both sides of the equation, we get:

$$(x + 4)^2 = 17 + 16$$

Simplifying the equation, we get:

$$(x + 4)^2 = 33$$

Taking the Square Root

To solve for $x$, we need to take the square root of both sides of the equation. This gives us:

$$x + 4 = \pm \sqrt{33}$$

Solving for $x$

Subtracting $4$ from both sides of the equation, we get:

$$x = -4 \pm \sqrt{33}$$

Conclusion

In this article, we have seen how to complete the square in the equation $x^2 + 8x = 17$. We have determined that the number to be added to both sides of the equation is $16$, which is the square of the coefficient of the $x$ term divided by $4$. This allows us to rewrite the equation in a perfect square form, which can be easily solved. The final solution to the equation is $x = -4 \pm \sqrt{33}$.

Frequently Asked Questions

• What is completing the square? Completing the square is a method of solving quadratic equations by rewriting them in a perfect square form.
• What is the formula for completing the square? The formula for completing the square is: $x^2 + bx = (x + \frac{b}{2})^2 - (\frac{b}{2})^2$
• What is the constant of completion? The constant of completion is the term added to the equation to complete the square.
• How do I determine the number to be added to both sides of the equation? To determine the number to be added to both sides of the equation, you need to square the coefficient of the $x$ term divided by $4$.

References

• [1] "Completing the Square" by Math Open Reference
• [2] "Completing the Square" by Khan Academy
• [3] "Completing the Square" by Purplemath

Further Reading

• "Quadratic Equations" by Math Is Fun
• "Solving Quadratic Equations" by IXL
• "Completing the Square" by Wolfram Alpha
  

Introduction

Completing the square is a powerful technique used in algebra to solve quadratic equations. It involves manipulating the equation to express it in a perfect square form, which can be easily solved. In this article, we will answer some of the most frequently asked questions about completing the square.

Q&A

Q: What is completing the square?

A: Completing the square is a method of solving quadratic equations by rewriting them in a perfect square form. This involves adding and subtracting a constant term to the equation, which allows us to express it as a perfect square trinomial.

Q: What is the formula for completing the square?

A: The formula for completing the square is:

$$x^2 + bx = (x + \frac{b}{2})^2 - (\frac{b}{2})^2$$

where $b$ is the coefficient of the $x$ term in the quadratic equation.

Q: What is the constant of completion?

A: The constant of completion is the term added to the equation to complete the square. It is equal to the square of the coefficient of the $x$ term divided by $4$.

Q: How do I determine the number to be added to both sides of the equation?

A: To determine the number to be added to both sides of the equation, you need to square the coefficient of the $x$ term divided by $4$. This will give you the constant of completion, which you need to add 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 expressing the quadratic equation as a product of two binomials, while completing the square involves rewriting the equation in a perfect square form.

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

A: No, completing the square is not suitable for all quadratic equations. It is best used for equations that can be rewritten in a perfect square form. If the equation cannot be rewritten in a perfect square form, you may need to use other methods, such as factoring or the quadratic formula.

Q: How do I know if an equation can be completed to a perfect square?

A: An equation can be completed to a perfect square if it can be rewritten in the form $(x + a)^2 = b$. If the equation can be rewritten in this form, it can be completed to a perfect square.

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

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

• Not squaring the coefficient of the $x$ term correctly
• Not adding the constant of completion to both sides of the equation
• Not checking the equation for extraneous solutions

Q: How do I check for extraneous solutions when completing the square?

A: To check for extraneous solutions, you need to plug the solutions back into the original equation and check if they are true. If the solutions are not true, they are extraneous and should be discarded.

Conclusion

Completing the square is a powerful technique used in algebra to solve quadratic equations. It involves manipulating the equation to express it in a perfect square form, which can be easily solved. By understanding the formula for completing the square and the constant of completion, you can determine the number to be added to both sides of the equation and solve the quadratic equation.

Frequently Asked Questions

• What is completing the square?
• What is the formula for completing the square?
• What is the constant of completion?
• How do I determine the number to be added to both sides of the equation?
• What is the difference between completing the square and factoring?
• Can I use completing the square to solve all quadratic equations?
• How do I know if an equation can be completed to a perfect square?
• What are some common mistakes to avoid when completing the square?
• How do I check for extraneous solutions when completing the square?

References

• [1] "Completing the Square" by Math Open Reference
• [2] "Completing the Square" by Khan Academy
• [3] "Completing the Square" by Purplemath

Further Reading

• "Quadratic Equations" by Math Is Fun
• "Solving Quadratic Equations" by IXL
• "Completing the Square" by Wolfram Alpha