Solve The Equation X 2 + 6 X − 4 = 0 X^2 + 6x - 4 = 0 X 2 + 6 X − 4 = 0 By Completing The Square. Give Your Answers Correct To 2 Decimal Places.${ \begin{align*} (x + 3)^2 - 13 &= 0 \ (x + 3)^2 &= 13 \end{align*} }$

Introduction

Completing the square is a method used to solve quadratic equations of the form $ax^2 + bx + c = 0$. This method involves manipulating the equation to express it in the form $(x + d)^2 = e$. In this article, we will use the completing the square method to solve the quadratic equation $x^2 + 6x - 4 = 0$.

Step 1: Write the Equation in the Correct Form

The given equation is $x^2 + 6x - 4 = 0$. To complete the square, we need to write the equation in the form $x^2 + bx + c = 0$. We can do this by moving the constant term to the right-hand side of the equation.

$x^2 + 6x - 4 = 0$

$x^2 + 6x = 4$

Step 2: Add and Subtract the Square of Half the Coefficient of x

The coefficient of $x$ is 6. Half of 6 is 3. The square of 3 is 9. We add and subtract 9 to the left-hand side of the equation.

$x^2 + 6x + 9 - 9 = 4$

$(x^2 + 6x + 9) - 9 = 4$

Step 3: Write the Left-Hand Side as a Perfect Square

The expression $x^2 + 6x + 9$ is a perfect square. We can write it as $(x + 3)^2$.

$(x + 3)^2 - 9 = 4$

Step 4: Add 9 to Both Sides of the Equation

We add 9 to both sides of the equation to get rid of the negative term.

$(x + 3)^2 = 4 + 9$

$(x + 3)^2 = 13$

Step 5: Take the Square Root of Both Sides of the Equation

We take the square root of both sides of the equation to solve for $x$.

$x + 3 = \pm \sqrt{13}$

Step 6: Solve for x

We solve for $x$ by subtracting 3 from both sides of the equation.

$x = -3 \pm \sqrt{13}$

Discussion

Completing the square is a method used to solve quadratic equations of the form $ax^2 + bx + c = 0$. This method involves manipulating the equation to express it in the form $(x + d)^2 = e$. In this article, we used the completing the square method to solve the quadratic equation $x^2 + 6x - 4 = 0$. We wrote the equation in the correct form, added and subtracted the square of half the coefficient of $x$, wrote the left-hand side as a perfect square, added 9 to both sides of the equation, and took the square root of both sides of the equation to solve for $x$.

Conclusion

In this article, we used the completing the square method to solve the quadratic equation $x^2 + 6x - 4 = 0$. We wrote the equation in the correct form, added and subtracted the square of half the coefficient of $x$, wrote the left-hand side as a perfect square, added 9 to both sides of the equation, and took the square root of both sides of the equation to solve for $x$. The solutions to the equation are $x = -3 \pm \sqrt{13}$.

Solutions to the Equation

The solutions to the equation are $x = -3 + \sqrt{13}$ and $x = -3 - \sqrt{13}$.

Calculating the Solutions

We can calculate the solutions to the equation using a calculator.

$x = -3 + \sqrt{13} \approx -3 + 3.61 \approx 0.61$

$x = -3 - \sqrt{13} \approx -3 - 3.61 \approx -6.61$

Conclusion

In this article, we used the completing the square method to solve the quadratic equation $x^2 + 6x - 4 = 0$. We wrote the equation in the correct form, added and subtracted the square of half the coefficient of $x$, wrote the left-hand side as a perfect square, added 9 to both sides of the equation, and took the square root of both sides of the equation to solve for $x$. The solutions to the equation are $x = -3 + \sqrt{13}$ and $x = -3 - \sqrt{13}$.

Final Answer

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Introduction

In our previous article, we used the completing the square method to solve the quadratic equation $x^2 + 6x - 4 = 0$. 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 method used to solve quadratic equations of the form $ax^2 + bx + c = 0$. This method involves manipulating the equation to express it in the form $(x + d)^2 = e$.

Q: How do I know if I should use completing the square to solve a quadratic equation?

A: You should use completing the square to solve a quadratic equation if the equation is in the form $ax^2 + bx + c = 0$ and you want to express the equation in the form $(x + d)^2 = e$. Completing the square is a good method to use 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. Write the equation in the correct form.
2. Add and subtract the square of half the coefficient of $x$.
3. Write the left-hand side as a perfect square.
4. Add the same value to both sides of the equation.
5. Take the square root of both sides of the equation.

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 half of the coefficient of $x$ and then square it. For example, if the coefficient of $x$ is 6, half of 6 is 3, and the square of 3 is 9. You would add and subtract 9 to the left-hand side of the equation.

Q: What if I get a negative value when I take the square root of both sides of the equation?

A: If you get a negative value when you take the square root of both sides of the equation, you need to consider both the positive and negative square roots. For example, if you get $x = -3 \pm \sqrt{13}$, you need to consider both $x = -3 + \sqrt{13}$ and $x = -3 - \sqrt{13}$.

Q: Can I use completing the square to solve any quadratic equation?

A: No, you cannot use completing the square to solve any quadratic equation. Completing the square is a method that works best for quadratic equations that are not easily factorable. If the equation is easily factorable, you may want to use factoring instead.

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 correct form.
• Not adding and subtracting the square of half the coefficient of $x$ correctly.
• Not writing the left-hand side as a perfect square.
• Not adding the same value to both sides of the equation.
• Not taking the square root of both sides of the equation correctly.

Conclusion

In this article, we answered some frequently asked questions about solving quadratic equations by completing the square. We discussed the steps to complete the square, how to add and subtract the square of half the coefficient of $x$, and what to do if you get a negative value when you take the square root of both sides of the equation. We also discussed some common mistakes to avoid when completing the square.

Final Answer

The final answer is that completing the square is a useful method for solving quadratic equations that are not easily factorable. By following the steps to complete the square and avoiding common mistakes, you can solve quadratic equations with ease.