Solve The Equation By Completing The Square. Round To The Nearest Hundredth If Necessary.$x^2 + 2x = 17$A. 4, 4.24 B. 3.24, -5.24 C. 17, -19 D. -3.24, 5.24

Introduction

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. This method involves manipulating the equation to express it in a perfect square trinomial form, which can then be easily solved. In this article, we will explore how to solve the quadratic equation $x^2 + 2x = 17$ by completing the square.

What is 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 can be easily solved by taking the square root of both sides of the equation.

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:

$x^2 + 2x = 17$

We can rewrite this equation as:

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

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 2, so we need to add and subtract $(2/2)^2 = 1$:

$x^2 + 2x - 17 + 1 - 1 = 0$

This can be rewritten as:

$(x^2 + 2x + 1) - 17 - 1 = 0$

Step 3: Factor the Perfect Square Trinomial

The expression $x^2 + 2x + 1$ is a perfect square trinomial, which can be factored as:

$(x + 1)^2$

So, we can rewrite the equation as:

$(x + 1)^2 - 18 = 0$

Step 4: Add 18 to Both Sides

The final step is to add 18 to both sides of the equation to isolate the perfect square trinomial:

$(x + 1)^2 = 18$

Step 5: Take the Square Root of Both Sides

We can now take the square root of both sides of the equation to solve for $x$:

$x + 1 = \pm \sqrt{18}$

Step 6: Simplify the Square Root

The square root of 18 can be simplified as:

$\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$

So, we can rewrite the equation as:

$x + 1 = \pm 3\sqrt{2}$

Step 7: Solve for x

We can now solve for $x$ by subtracting 1 from both sides of the equation:

$x = -1 \pm 3\sqrt{2}$

Rounding to the Nearest Hundredth

To round the solutions to the nearest hundredth, we can evaluate the expression $-1 \pm 3\sqrt{2}$:

$x \approx -1 \pm 4.24$

So, the two solutions are:

$x \approx 3.24$ and $x \approx -5.24$

Conclusion

In this article, we have shown how to solve the quadratic equation $x^2 + 2x = 17$ 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 18 to both sides, taking the square root of both sides, and simplifying the square root. We have also rounded the solutions to the nearest hundredth. The two solutions are $x \approx 3.24$ and $x \approx -5.24$.

Answer

The correct answer is:

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 trinomial form, which can then be easily solved. In this article, we will answer some frequently asked questions about 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 to complete the square?

A: The steps to complete 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 the value that was subtracted in step 2 to both sides of the equation.
5. Take the square root of both sides of the equation.
6. Simplify the square root.

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 find the value of half the coefficient of $x$ and then square it. For example, if the coefficient of $x$ is 2, you would add and subtract $(2/2)^2 = 1$.

Q: What if the equation has a coefficient of $x^2$ that is not 1?

A: If the equation has a coefficient of $x^2$ that is not 1, you need to factor out the coefficient of $x^2$ before completing the square. For example, if the equation is $2x^2 + 4x + 3 = 0$, you would first factor out the coefficient of $x^2$ to get $2(x^2 + 2x) + 3 = 0$.

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

A: Yes, you can use completing the square to solve quadratic equations with complex solutions. However, you need to be careful when taking the square root of both sides of the equation, as this can introduce complex solutions.

Q: How do I know if my solution is correct?

A: To check if your solution is correct, you can plug it back into the original equation and see if it satisfies the equation. If it does, then your solution is correct.

Conclusion

In this article, we have answered some frequently asked questions about solving quadratic equations by completing the square. We have covered topics such as what completing the square is, when to use it, the steps to complete the square, and how to add and subtract the square of half the coefficient of $x$. We have also discussed how to handle equations with coefficients of $x^2$ that are not 1 and how to solve quadratic equations with complex solutions.

Common Mistakes to Avoid

When solving quadratic equations by completing the square, there are several common mistakes to avoid. These 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 the value that was subtracted in step 2 to both sides of the equation
• Not taking the square root of both sides of the equation
• Not simplifying the square root

By avoiding these common mistakes, you can ensure that your solution is correct and that you have successfully completed the square.

Practice Problems

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

1. Solve the equation $x^2 + 4x + 4 = 0$ by completing the square.
2. Solve the equation $2x^2 + 6x + 5 = 0$ by completing the square.
3. Solve the equation $x^2 - 6x + 9 = 0$ by completing the square.

We hope that this article has been helpful in answering your questions about solving quadratic equations by completing the square. If you have any further questions, please don't hesitate to ask.