Complete The Square To Solve The Equation Below:$\[ X^2 + X = \frac{7}{4} \\]A. $\[ X = 1 + \sqrt{2}; \quad X = 1 - \sqrt{2} \\]B. $\[ X = -\frac{1}{2} + \sqrt{2}; \quad X = -\frac{1}{2} - \sqrt{2} \\]C. $\[ X = -2 +

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. One of the methods used to solve quadratic equations is by completing the square. This method involves manipulating the equation to express it in a perfect square form, which can then be easily solved. In this article, we will explore how to complete the square to solve the equation $x^2 + x = \frac{7}{4}$.

What is Completing the Square?

Completing the square is a technique used to solve quadratic equations by expressing them in a perfect square form. This involves adding and subtracting a constant term to the equation, which allows us to rewrite it in a form that can be easily solved. The constant term added is called the "square root of the constant term" and is calculated by taking half of the coefficient of the linear term and squaring it.

Step-by-Step Guide to Completing the Square

To complete the square, we will follow these steps:

1. Write the equation in the standard form: The equation should be written in the standard form $ax^2 + bx + c = 0$.
2. Identify the coefficient of the linear term: The coefficient of the linear term is the number that multiplies the variable $x$.
3. Calculate the square root of the constant term: The square root of the constant term is calculated by taking half of the coefficient of the linear term and squaring it.
4. Add and subtract the square root of the constant term: Add and subtract the square root of the constant term to the equation.
5. Rewrite the equation in a perfect square form: The equation can now be rewritten in a perfect square form.

Solving the Equation $x^2 + x = \frac{7}{4}$

Now that we have understood the steps involved in completing the square, let's apply them to the equation $x^2 + x = \frac{7}{4}$.

Step 1: Write the equation in the standard form

The equation is already in the standard form $x^2 + x = \frac{7}{4}$.

Step 2: Identify the coefficient of the linear term

The coefficient of the linear term is 1.

Step 3: Calculate the square root of the constant term

The square root of the constant term is calculated by taking half of the coefficient of the linear term and squaring it. In this case, the coefficient of the linear term is 1, so the square root of the constant term is $\frac{1}{2}^2 = \frac{1}{4}$.

Step 4: Add and subtract the square root of the constant term

Add and subtract the square root of the constant term to the equation:

$x^2 + x + \frac{1}{4} - \frac{1}{4} = \frac{7}{4}$

Step 5: Rewrite the equation in a perfect square form

The equation can now be rewritten in a perfect square form:

$(x + \frac{1}{2})^2 - \frac{1}{4} = \frac{7}{4}$

Simplifying the Equation

Now that we have rewritten the equation in a perfect square form, let's simplify it:

$(x + \frac{1}{2})^2 = \frac{7}{4} + \frac{1}{4}$

$(x + \frac{1}{2})^2 = \frac{8}{4}$

$(x + \frac{1}{2})^2 = 2$

Taking the Square Root

Now that we have simplified the equation, let's take the square root of both sides:

$x + \frac{1}{2} = \pm \sqrt{2}$

Solving for x

Now that we have taken the square root of both sides, let's solve for x:

$x = -\frac{1}{2} \pm \sqrt{2}$

Conclusion

In this article, we have learned how to complete the square to solve the equation $x^2 + x = \frac{7}{4}$. We have followed the steps involved in completing the square, including writing the equation in the standard form, identifying the coefficient of the linear term, calculating the square root of the constant term, adding and subtracting the square root of the constant term, and rewriting the equation in a perfect square form. We have also simplified the equation and taken the square root of both sides to solve for x. The solutions to the equation are $x = -\frac{1}{2} + \sqrt{2}$ and $x = -\frac{1}{2} - \sqrt{2}$.

Final Answer

The final answer is:

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Introduction

Completing the square is a powerful technique used to solve quadratic equations. In our previous article, we explored how to complete the square to solve the equation $x^2 + x = \frac{7}{4}$. In this article, we will answer some of the most frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a technique used to solve quadratic equations by expressing them in a perfect square form. This involves adding and subtracting a constant term to the equation, which allows us to rewrite it in a form that can be easily solved.

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

A: You should use completing the square when the quadratic equation is in the form $ax^2 + bx + c = 0$ and you want to find the solutions in the form $(x + d)^2 = e$. Completing the square is particularly useful when the quadratic equation has a linear term, as it allows you to rewrite the equation in a perfect square form.

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

A: The steps involved in completing the square are:

1. Write the equation in the standard form: The equation should be written in the standard form $ax^2 + bx + c = 0$.
2. Identify the coefficient of the linear term: The coefficient of the linear term is the number that multiplies the variable $x$.
3. Calculate the square root of the constant term: The square root of the constant term is calculated by taking half of the coefficient of the linear term and squaring it.
4. Add and subtract the square root of the constant term: Add and subtract the square root of the constant term to the equation.
5. Rewrite the equation in a perfect square form: The equation can now be rewritten in a perfect square form.

Q: How do I calculate the square root of the constant term?

A: To calculate the square root of the constant term, you need to take half of the coefficient of the linear term and square it. For example, if the coefficient of the linear term is 1, the square root of the constant term is $\frac{1}{2}^2 = \frac{1}{4}$.

Q: What if the quadratic equation has a negative coefficient of the linear term?

A: If the quadratic equation has a negative coefficient of the linear term, you need to add and subtract the square root of the constant term in the opposite order. For example, if the quadratic equation is $x^2 - x = \frac{7}{4}$, you would add and subtract the square root of the constant term as follows:

$x^2 - x + \frac{1}{4} - \frac{1}{4} = \frac{7}{4}$

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

A: Yes, you can use completing the square to solve quadratic equations with complex coefficients. The steps involved in completing the square remain the same, but you need to be careful when working with complex numbers.

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

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

• Not writing the equation in the standard form: Make sure to write the equation in the standard form $ax^2 + bx + c = 0$ before attempting to complete the square.
• Not identifying the coefficient of the linear term: Make sure to identify the coefficient of the linear term correctly before calculating the square root of the constant term.
• Not adding and subtracting the square root of the constant term correctly: Make sure to add and subtract the square root of the constant term in the correct order.
• Not rewriting the equation in a perfect square form: Make sure to rewrite the equation in a perfect square form after completing the square.

Conclusion

In this article, we have answered some of the most frequently asked questions about completing the square. We have covered the steps involved in completing the square, how to calculate the square root of the constant term, and some common mistakes to avoid. By following these steps and avoiding common mistakes, you can use completing the square to solve quadratic equations with ease.

Final Tips

• Practice, practice, practice: The best way to learn completing the square is by practicing it. Try solving different quadratic equations using completing the square.
• Use online resources: There are many online resources available that can help you learn completing the square, including video tutorials and practice problems.
• Seek help when needed: If you are struggling with completing the square, don't hesitate to seek help from a teacher or tutor.