Type The Correct Answer In Each Box. Use Numerals Instead Of Words.Consider The Quadratic Equation $x^2 + 10x + 27 = 0$.Completing The Square Leads To The Equivalent Equation $(x + \square)^2 = \square$.

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Introduction

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Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. One method of solving quadratic equations is by completing the square, which involves manipulating the equation to express it in a perfect square form. In this article, we will explore how to complete the square for the quadratic equation $x^2 + 10x + 27 = 0$.

Understanding the Quadratic Equation

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A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In our example, the quadratic equation is $x^2 + 10x + 27 = 0$, where $a = 1$, $b = 10$, and $c = 27$.

Completing the Square

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Completing the square is a method of solving quadratic equations by expressing them in a perfect square form. This involves adding and subtracting a constant term to create a perfect square trinomial. The general form of a perfect square trinomial is $(x + d)^2 = x^2 + 2dx + d^2$, where $d$ is a constant.

To complete the square for the quadratic equation $x^2 + 10x + 27 = 0$, we need to find the value of $d$ that makes the equation a perfect square trinomial. We can do this by taking half of the coefficient of the $x$ term and squaring it. In this case, the coefficient of the $x$ term is $10$, so we take half of $10$ and square it to get $5^2 = 25$.

Finding the Value of $d$

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Now that we have the value of $d$, we can substitute it into the perfect square trinomial form to get $(x + 5)^2 = x^2 + 10x + 25$. We can see that this is equivalent to the original quadratic equation $x^2 + 10x + 27 = 0$, except for the constant term.

Solving for $x$

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To solve for $x$, we can set the perfect square trinomial equal to zero and solve for $x$. We have $(x + 5)^2 = 25$, which can be rewritten as $x + 5 = \pm \sqrt{25}$. Taking the square root of both sides, we get $x + 5 = \pm 5$.

Solving for $x$ (continued)

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Now that we have the two possible values of $x + 5$, we can solve for $x$ by subtracting $5$ from both sides. We have $x + 5 = 5$ and $x + 5 = -5$. Subtracting $5$ from both sides of the first equation, we get $x = 0$. Subtracting $5$ from both sides of the second equation, we get $x = -10$.

Conclusion

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In this article, we have seen how to complete the square for the quadratic equation $x^2 + 10x + 27 = 0$. We have found the value of $d$ that makes the equation a perfect square trinomial, and we have solved for $x$ by setting the perfect square trinomial equal to zero and solving for $x$. The two possible values of $x$ are $x = 0$ and $x = -10$.

Final Answer

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The final answer is:

• x = 0
• x = -10

Note: The final answer is in the format of a list, with each item representing a possible value of $x$.

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Introduction

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In our previous article, we explored how to complete the square for the quadratic equation $x^2 + 10x + 27 = 0$. Completing the square is a powerful method for solving quadratic equations, and it has many applications in mathematics and science. In this article, we will answer some common questions about completing the square and provide additional examples to help you understand the concept.

Q: What is completing the square?

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A: Completing the square is a method of solving quadratic equations by expressing them in a perfect square form. This involves adding and subtracting a constant term to create a perfect square trinomial.

Q: How do I know when to use completing the square?

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A: You should use completing the square when the quadratic equation is in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. Completing the square is particularly useful when the equation is not easily factorable.

Q: What is the formula for completing the square?

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A: The formula for completing the square is:

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

This formula allows you to rewrite the quadratic equation in a perfect square form.

Q: How do I find the value of $d$?

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A: To find the value of $d$, you need to take half of the coefficient of the $x$ term and square it. In the formula above, $d$ is equal to $\frac{b}{2}$.

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 adding and subtracting the same constant term
• Not squaring the value of $d$
• Not setting the perfect square trinomial equal to zero

Q: Can I use completing the square to solve quadratic equations with complex coefficients?

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A: Yes, you can use completing the square to solve quadratic equations with complex coefficients. However, you will need to use complex numbers and follow the same steps as before.

Q: Are there any other methods for solving quadratic equations?

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A: Yes, there are several other methods for solving quadratic equations, including:

• Factoring
• Using the quadratic formula
• Graphing

Each method has its own advantages and disadvantages, and the choice of method will depend on the specific equation and the level of difficulty.

Q: Can I use completing the square to solve quadratic equations with rational coefficients?

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A: Yes, you can use completing the square to solve quadratic equations with rational coefficients. However, you will need to follow the same steps as before and be careful when working with rational numbers.

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

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

• Physics: Completing the square is used to solve equations of motion and to find the position and velocity of an object.
• Engineering: Completing the square is used to design and optimize systems, such as electrical circuits and mechanical systems.
• Computer Science: Completing the square is used in algorithms for solving systems of linear equations and for finding the shortest path between two points.

Conclusion

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In this article, we have answered some common questions about completing the square and provided additional examples to help you understand the concept. Completing the square is a powerful method for solving quadratic equations, and it has many applications in mathematics and science. With practice and patience, you can master the art of completing the square and solve quadratic equations with ease.

Final Answer

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The final answer is:

• Completing the square is a method of solving quadratic equations by expressing them in a perfect square form.
• The formula for completing the square is: $(x + \frac{b}{2})^2 = (\frac{b}{2})^2 - c$
• To find the value of $d$, you need to take half of the coefficient of the $x$ term and square it.