Complete The Square To Solve The Equation Below.$\[ X^2 + 10x - 13 = 17 \\]A. \[$ X = -10 + \sqrt{55} ; X = -10 - \sqrt{55} \$\]B. \[$ X = 4 + \sqrt{30} ; X = 4 - \sqrt{30} \$\]C. \[$ X = -5 + \sqrt{55} ; X = -5 - \sqrt{55}

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Introduction

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In algebra, completing the square is a powerful technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square trinomial form, which can then be easily solved. In this article, we will learn how to complete the square to solve the equation $x^2 + 10x - 13 = 17$. We will also explore the different methods of completing the square and provide step-by-step solutions to the given equation.

What is Completing the Square?

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Completing the square is a method of solving quadratic equations by manipulating the equation to express it in a perfect square trinomial form. This involves adding and subtracting a constant term to create a perfect square trinomial, which can then be easily solved. The process of completing the square involves the following steps:

1. Write the equation in the form $ax^2 + bx + c = 0$: The first step is to write the equation in the standard form $ax^2 + bx + c = 0$.
2. Move the constant term to the right-hand side: The next step is to move the constant term to the right-hand side of the equation.
3. Add and subtract $(b/2)^2$: The next step is to add and subtract $(b/2)^2$ to the left-hand side of the equation.
4. Factor the perfect square trinomial: The final step is to factor the perfect square trinomial on the left-hand side of the equation.

Step-by-Step Solution to the Equation

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Now that we have learned the steps involved in completing the square, let's apply them to the given equation $x^2 + 10x - 13 = 17$.

Step 1: Write the equation in the form $ax^2 + bx + c = 0$

The given equation is $x^2 + 10x - 13 = 17$. To write it in the standard form $ax^2 + bx + c = 0$, we need to subtract 17 from both sides of the equation.

$x^2 + 10x - 13 - 17 = 0$

$x^2 + 10x - 30 = 0$

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

The equation is already in the standard form $ax^2 + bx + c = 0$, so we don't need to do anything here.

Step 3: Add and subtract $(b/2)^2$

The coefficient of $x$ is 10, so we need to add and subtract $(10/2)^2 = 25$ to the left-hand side of the equation.

$x^2 + 10x + 25 - 25 - 30 = 0$

Step 4: Factor the perfect square trinomial

The left-hand side of the equation is now a perfect square trinomial, which can be factored as follows:

$(x + 5)^2 - 55 = 0$

Solving the Equation

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Now that we have factored the perfect square trinomial, we can solve the equation by setting each factor equal to zero.

$(x + 5)^2 = 55$

$x + 5 = \pm \sqrt{55}$

$x = -5 \pm \sqrt{55}$

Therefore, the solutions to the equation are $x = -5 + \sqrt{55}$ and $x = -5 - \sqrt{55}$.

Conclusion

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In this article, we learned how to complete the square to solve the equation $x^2 + 10x - 13 = 17$. We applied the steps involved in completing the square to the given equation and obtained the solutions $x = -5 + \sqrt{55}$ and $x = -5 - \sqrt{55}$. We also explored the different methods of completing the square and provided step-by-step solutions to the given equation.

Final Answer

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The final answer to the equation $x^2 + 10x - 13 = 17$ is:

$x = -5 + \sqrt{55}$ and $x = -5 - \sqrt{55}$

This is option C.

Comparison of Options

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Let's compare the solutions we obtained with the options given:

Option A: $x = -10 + \sqrt{55}$ and $x = -10 - \sqrt{55}$

Option B: $x = 4 + \sqrt{30}$ and $x = 4 - \sqrt{30}$

Option C: $x = -5 + \sqrt{55}$ and $x = -5 - \sqrt{55}$

Our solutions match option C.

Conclusion

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In conclusion, we have successfully completed the square to solve the equation $x^2 + 10x - 13 = 17$. We obtained the solutions $x = -5 + \sqrt{55}$ and $x = -5 - \sqrt{55}$, which match option C.

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Introduction

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Completing the square is a powerful technique used to solve quadratic equations. In our previous article, we learned how to complete the square to solve the equation $x^2 + 10x - 13 = 17$. In this article, we will provide a Q&A guide to help you understand the concept of completing the square and how to apply it to solve quadratic equations.

Q: What is completing the square?

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A: Completing the square is a method of solving quadratic equations by manipulating the equation to express it in a perfect square trinomial form. This involves adding and subtracting a constant term to create a perfect square trinomial, which can then be easily solved.

Q: What are the steps involved in completing the square?

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A: The steps involved in completing the square are:

1. Write the equation in the form $ax^2 + bx + c = 0$: The first step is to write the equation in the standard form $ax^2 + bx + c = 0$.
2. Move the constant term to the right-hand side: The next step is to move the constant term to the right-hand side of the equation.
3. Add and subtract $(b/2)^2$: The next step is to add and subtract $(b/2)^2$ to the left-hand side of the equation.
4. Factor the perfect square trinomial: The final step is to factor the perfect square trinomial on the left-hand side of the equation.

Q: How do I know when to add and subtract $(b/2)^2$?

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A: You need to add and subtract $(b/2)^2$ when the coefficient of $x$ is not zero. The coefficient of $x$ is the number that is multiplied by $x$ in the equation.

Q: What is the purpose of adding and subtracting $(b/2)^2$?

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A: The purpose of adding and subtracting $(b/2)^2$ is to create a perfect square trinomial on the left-hand side of the equation. This allows you to factor the equation and solve for the variable.

Q: How do I factor the perfect square trinomial?

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A: To factor the perfect square trinomial, you need to identify the binomial that is being squared. The binomial is the expression that is being squared to create the perfect square trinomial.

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

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

• Not moving the constant term to the right-hand side: Make sure to move the constant term to the right-hand side of the equation.
• Not adding and subtracting $(b/2)^2$: Make sure to add and subtract $(b/2)^2$ when the coefficient of $x$ is not zero.
• Not factoring the perfect square trinomial: Make sure to factor the perfect square trinomial on the left-hand side of the equation.

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

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A: To check your work when completing the square, you need to plug the solutions back into the original equation to make sure they are true. This will help you verify that your solutions are correct.

Conclusion

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In conclusion, completing the square is a powerful technique used to solve quadratic equations. By following the steps involved in completing the square and avoiding common mistakes, you can successfully solve quadratic equations and check your work. We hope this Q&A guide has helped you understand the concept of completing the square and how to apply it to solve quadratic equations.

Final Tips

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• Practice, practice, practice: The more you practice completing the square, the more comfortable you will become with the process.
• Use online resources: There are many online resources available that can help you learn and practice completing the square.
• Seek help when needed: Don't be afraid to ask for help if you are struggling with completing the square.