Complete The Square To Solve 2 X 2 + 20 X = 10 2x^2 + 20x = 10 2 X 2 + 20 X = 10 .A. X = − 2 ± 10 X = -2 \pm \sqrt{10} X = − 2 ± 10 ​ B. X = − 5 ± 30 X = -5 \pm \sqrt{30} X = − 5 ± 30 ​ C. X = 2 ± 15 X = 2 \pm \sqrt{15} X = 2 ± 15 ​ D. X = 5 ± 30 X = 5 \pm \sqrt{30} X = 5 ± 30 ​

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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 and professionals alike. One of the most effective methods for solving quadratic equations is the method of completing the square. In this article, we will explore the concept of completing the square and apply it to solve the quadratic equation $2x^2 + 20x = 10$.

What is Completing the Square?

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Completing the square is a technique used to solve quadratic equations by rewriting them in a perfect square form. This involves manipulating the equation to create a perfect square trinomial on one side of the equation, which can then be factored into the square of a binomial. The method of completing the square is based on the concept of the square of a binomial, which is a fundamental concept in algebra.

The Square of a Binomial

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The square of a binomial is a fundamental concept in algebra, and it is used extensively in the method of completing the square. The square of a binomial is given by the formula:

$(a + b)^2 = a^2 + 2ab + b^2$

This formula can be used to expand the square of a binomial, which is a crucial step in completing the square.

How to Complete the Square

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Completing the square involves several steps, which are outlined below:

1. Write the quadratic equation in the form $ax^2 + bx + c = 0$: The first step in completing the square is to write the quadratic equation in the form $ax^2 + bx + c = 0$. This will allow us to identify the coefficients of the quadratic equation.
2. Divide the equation by the coefficient of $x^2$: The next step is to divide the equation by the coefficient of $x^2$. This will ensure that the coefficient of $x^2$ is equal to 1, which is a necessary condition for completing the square.
3. 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. This will allow us to create a perfect square trinomial on the left-hand side of the equation.
4. Add and subtract the square of half the coefficient of $x$: The next step is to add and subtract the square of half the coefficient of $x$ to the left-hand side of the equation. This will create a perfect square trinomial on the left-hand side of the equation.
5. Factor the perfect square trinomial: The final step is to factor the perfect square trinomial on the left-hand side of the equation. This will allow us to solve for $x$.

Solving the Quadratic Equation $2x^2 + 20x = 10$

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Now that we have outlined the steps for completing the square, let's apply them to solve the quadratic equation $2x^2 + 20x = 10$.

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

The first step is to write the quadratic equation in the form $ax^2 + bx + c = 0$. We can do this by subtracting 10 from both sides of the equation:

$2x^2 + 20x - 10 = 0$

Step 2: Divide the equation by the coefficient of $x^2$

The next step is to divide the equation by the coefficient of $x^2$. In this case, the coefficient of $x^2$ is 2, so we can divide the equation by 2:

$x^2 + 10x - 5 = 0$

Step 3: 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. We can do this by adding 5 to both sides of the equation:

$x^2 + 10x = 5$

Step 4: Add and subtract the square of half the coefficient of $x$

The next step is to add and subtract the square of half the coefficient of $x$ to the left-hand side of the equation. In this case, the coefficient of $x$ is 10, so we can add and subtract the square of half of 10, which is 25:

$x^2 + 10x + 25 = 5 + 25$

Step 5: Factor the perfect square trinomial

The final step is to factor the perfect square trinomial on the left-hand side of the equation. We can do this by factoring the square of the binomial $(x + 5)$:

$(x + 5)^2 = 30$

Solving for $x$

Now that we have factored the perfect square trinomial, we can solve for $x$ by taking the square root of both sides of the equation:

$x + 5 = \pm \sqrt{30}$

Subtracting 5 from both sides of the equation gives us:

$x = -5 \pm \sqrt{30}$

Conclusion

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In this article, we have explored the concept of completing the square and applied it to solve the quadratic equation $2x^2 + 20x = 10$. We have outlined the steps for completing the square and used them to solve the quadratic equation. The final solution is $x = -5 \pm \sqrt{30}$.

Answer

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The correct answer is:

B. $x = -5 \pm \sqrt{30}$

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Introduction

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Completing the square is a powerful technique for solving quadratic equations. In our previous article, we explored the concept of completing the square and applied it to solve the quadratic equation $2x^2 + 20x = 10$. In this article, we will answer some of the most frequently asked questions about completing the square.

Q: What is completing the square?

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A: Completing the square is a technique used to solve quadratic equations by rewriting them in a perfect square form. This involves manipulating the equation to create a perfect square trinomial on one side of the equation, which can then be factored into the square of a binomial.

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

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A: You should use completing the square when you are given a quadratic equation in the form $ax^2 + bx + c = 0$ and you want to solve for $x$. Completing the square is a useful technique when the quadratic equation cannot be factored easily.

Q: What are the steps for completing the square?

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A: The steps for completing the square are:

1. Write the quadratic equation in the form $ax^2 + bx + c = 0$.
2. Divide the equation by the coefficient of $x^2$.
3. Move the constant term to the right-hand side.
4. Add and subtract the square of half the coefficient of $x$.
5. Factor the perfect square trinomial.

Q: How do I add and subtract the square of half the coefficient of $x$?

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A: To add and subtract the square of half the coefficient of $x$, you need to find the square of half of the coefficient of $x$. For example, if the coefficient of $x$ is 10, you would add and subtract the square of half of 10, which is 25.

Q: What is the difference between completing the square and factoring?

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A: Completing the square and factoring are two different techniques for solving quadratic equations. Factoring involves finding two binomials whose product is equal to the quadratic expression, while completing the square involves rewriting the quadratic expression in a perfect square form.

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

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A: No, you cannot use completing the square to solve all quadratic equations. Completing the square is a useful technique when the quadratic equation cannot be factored easily, but it may not be the best technique for all quadratic equations.

Q: How do I know if a quadratic equation can be solved using completing the square?

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A: You can determine if a quadratic equation can be solved using completing the square by checking if the equation can be written in the form $ax^2 + bx + c = 0$. If the equation can be written in this form, you can try completing the square to solve it.

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 dividing the equation by the coefficient of $x^2$.
• Not moving the constant term to the right-hand side.
• Not adding and subtracting the square of half the coefficient of $x$.
• Not factoring the perfect square trinomial.

Conclusion

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In this article, we have answered some of the most frequently asked questions about completing the square. We have discussed the concept of completing the square, the steps for completing the square, and some common mistakes to avoid. By following these steps and avoiding these mistakes, you can use completing the square to solve quadratic equations with ease.

Additional Resources

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If you want to learn more about completing the square, we recommend checking out the following resources:

• Khan Academy: Completing the Square
• Mathway: Completing the Square
• Wolfram Alpha: Completing the Square

Practice Problems

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To practice completing the square, try solving the following quadratic equations:

• $x^2 + 6x + 8 = 0$
• $x^2 - 4x + 4 = 0$
• $x^2 + 2x + 1 = 0$

By practicing completing the square, you can become more confident and proficient in using this technique to solve quadratic equations.