Solve The Equation For All Values Of $x$ By Completing The Square. Express Your Answer In Simplest Form.$\[ X^2 + 4x = 5 \\]$x = \square$(Note: Show Your Work And Submit Your Answer.)

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. One of the methods used to solve quadratic equations is 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 solve the quadratic equation $x^2 + 4x = 5$ by completing the square.

Understanding the Method of Completing the Square

Completing the square is a technique used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The method involves manipulating the equation to express it in the form $(x + d)^2 = e$, where $d$ and $e$ are constants. This form is called a perfect square trinomial, and it can be easily solved by taking the square root of both sides.

Step 1: Move the Constant Term to the Right-Hand Side

The first step in completing the square is to move the constant term to the right-hand side of the equation. In this case, we have:

$x^2 + 4x = 5$

We can rewrite this equation by subtracting 5 from both sides:

$x^2 + 4x - 5 = 0$

Step 2: 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 4, so we need to add and subtract $(4/2)^2 = 4$:

$x^2 + 4x - 5 + 4 - 4 = 0$

This can be rewritten as:

$x^2 + 4x - 4 - 5 + 4 = 0$

Step 3: Factor the Perfect Square Trinomial

Now that we have added and subtracted the square of half the coefficient of $x$, we can factor the perfect square trinomial:

$(x + 2)^2 - 9 = 0$

Step 4: Add 9 to Both Sides

The final step is to add 9 to both sides of the equation to isolate the perfect square trinomial:

$(x + 2)^2 = 9$

Step 5: Take the Square Root of Both Sides

Now that we have isolated the perfect square trinomial, we can take the square root of both sides of the equation:

$x + 2 = \pm \sqrt{9}$

Step 6: Simplify the Equation

The final step is to simplify the equation by evaluating the square root:

$x + 2 = \pm 3$

Step 7: Solve for $x$

Now that we have simplified the equation, we can solve for $x$ by subtracting 2 from both sides:

$x = -2 \pm 3$

Simplifying the Solution

The final step is to simplify the solution by combining the two possible values of $x$:

$x = -2 + 3$ or $x = -2 - 3$

$x = 1$ or $x = -5$

Conclusion

In this article, we have explored how to solve the quadratic equation $x^2 + 4x = 5$ by completing the square. We have followed the steps of moving the constant term to the right-hand side, adding and subtracting the square of half the coefficient of $x$, factoring the perfect square trinomial, adding 9 to both sides, taking the square root of both sides, simplifying the equation, and solving for $x$. The final solution is $x = 1$ or $x = -5$.

Final Answer

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Introduction

In our previous article, we explored how to solve quadratic equations 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 answer some common questions related to solving quadratic equations by completing the square.

Q: What is completing the square?

A: Completing the square is a technique used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The method involves manipulating the equation to express it in the form $(x + d)^2 = e$, where $d$ and $e$ are constants.

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 $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. This method is particularly useful when the equation is not easily factorable.

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

A: The steps involved in completing the square are:

1. Move the constant term to the right-hand side of the equation.
2. Add and subtract the square of half the coefficient of $x$ to the left-hand side of the equation.
3. Factor the perfect square trinomial.
4. Add 9 to both sides of the equation.
5. Take the square root of both sides of the equation.
6. Simplify the equation.
7. Solve for $x$.

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

A: To add and subtract the square of half the coefficient of $x$, you need to add and subtract $(b/2)^2$ to the left-hand side of the equation. For example, if the coefficient of $x$ is 4, you would add and subtract $(4/2)^2 = 4$.

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

A: Completing the square and factoring are two different methods used to solve quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while completing the square involves manipulating the equation to express it in a perfect square form.

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

A: No, you cannot use completing the square to solve all quadratic equations. This method is particularly useful when the equation is not easily factorable, but it may not be the best method to use when the equation is easily factorable.

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

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

• Not moving the constant term to the right-hand side of the equation.
• Not adding and subtracting the square of half the coefficient of $x$.
• Not factoring the perfect square trinomial.
• Not adding 9 to both sides of the equation.
• Not taking the square root of both sides of the equation.

Conclusion

In this article, we have answered some common questions related to solving quadratic equations by completing the square. We have discussed the steps involved in completing the square, how to add and subtract the square of half the coefficient of $x$, and some common mistakes to avoid. By following these steps and avoiding common mistakes, you can use completing the square to solve quadratic equations.

Final Answer

The final answer is $\boxed{1, -5}$.