Solve $x^2 - 8x - 9 = 0$.1. Rewrite The Equation In The Form $x^2 + Bx = C$: - $x^2 - 8x = 9$2. Complete The Square: - Add $\square$ To Each Side Of $x^2 - 8x = 9$ To Complete The Square.

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Introduction

In this article, we will solve the quadratic equation $x^2 - 8x - 9 = 0$ using the method of completing the square. This method is a powerful tool for solving quadratic equations and is often used in mathematics and physics.

Step 1: Rewrite the equation in the form $x^2 + bx = c$

To begin, we need to rewrite the equation in the form $x^2 + bx = c$. We can do this by adding $9$ to both sides of the equation:

$x^2 - 8x - 9 = 0$

$x^2 - 8x = 9$

Now, we have the equation in the form $x^2 + bx = c$, where $b = -8$ and $c = 9$.

Step 2: Complete the square

To complete the square, we need to add $\square$ to each side of the equation. The value of $\square$ is half of the coefficient of the $x$ term, squared. In this case, the coefficient of the $x$ term is $-8$, so we need to add $\left(\frac{-8}{2}\right)^2 = 16$ to each side of the equation:

$x^2 - 8x = 9$

$x^2 - 8x + 16 = 9 + 16$

Now, we can rewrite the left-hand side of the equation as a perfect square:

$x^2 - 8x + 16 = (x - 4)^2$

So, we have:

$(x - 4)^2 = 25$

Discussion

Now that we have completed the square, we can solve for $x$. We can take the square root of both sides of the equation to get:

$x - 4 = \pm \sqrt{25}$

$x - 4 = \pm 5$

Now, we can add $4$ to both sides of the equation to get:

$x = 4 \pm 5$

So, we have two possible solutions for $x$:

$x = 9$ or $x = -1$

Conclusion

In this article, we solved the quadratic equation $x^2 - 8x - 9 = 0$ using the method of completing the square. We first rewrote the equation in the form $x^2 + bx = c$, and then completed the square by adding $\square$ to each side of the equation. Finally, we solved for $x$ by taking the square root of both sides of the equation. The two possible solutions for $x$ are $x = 9$ and $x = -1$.

Additional Examples

Here are a few more examples of quadratic equations that can be solved using the method of completing the square:

• $x^2 + 6x - 8 = 0$
• $x^2 - 2x - 15 = 0$
• $x^2 + 4x - 5 = 0$

Step-by-Step Solution

Here is a step-by-step solution to the quadratic equation $x^2 - 8x - 9 = 0$:

1. Rewrite the equation in the form $x^2 + bx = c$ by adding $9$ to both sides of the equation:

   $x^2 - 8x - 9 = 0$

   $x^2 - 8x = 9$

2. Complete the square by adding $\square$ to each side of the equation. The value of $\square$ is half of the coefficient of the $x$ term, squared. In this case, the coefficient of the $x$ term is $-8$, so we need to add $\left(\frac{-8}{2}\right)^2 = 16$ to each side of the equation:

   $x^2 - 8x = 9$

   $x^2 - 8x + 16 = 9 + 16$

3. Rewrite the left-hand side of the equation as a perfect square:

   $x^2 - 8x + 16 = (x - 4)^2$

   So, we have:

   $(x - 4)^2 = 25$

4. Take the square root of both sides of the equation to get:

   $x - 4 = \pm \sqrt{25}$

   $x - 4 = \pm 5$

5. Add $4$ to both sides of the equation to get:

   $x = 4 \pm 5$

   So, we have two possible solutions for $x$:

   $x = 9$ or $x = -1$

Mathematical Formulas

Here are some mathematical formulas that are used in the solution to the quadratic equation $x^2 - 8x - 9 = 0$:

• $(x - a)^2 = x^2 - 2ax + a^2$
• $\sqrt{a^2} = a$

Conclusion

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Q: What is the method of completing the square?

A: The method of completing the square is a technique used to solve quadratic equations by rewriting them in a form that allows us to easily find the solutions. It involves adding a constant term to both sides of the equation to create a perfect square trinomial.

Q: How do I rewrite the equation in the form $x^2 + bx = c$?

A: To rewrite the equation in the form $x^2 + bx = c$, you need to add the constant term to both sides of the equation. For example, if you have the equation $x^2 - 8x - 9 = 0$, you would add $9$ to both sides to get $x^2 - 8x = 9$.

Q: What is the value of $\square$ that I need to add to each side of the equation?

A: The value of $\square$ is half of the coefficient of the $x$ term, squared. In the case of the equation $x^2 - 8x = 9$, the coefficient of the $x$ term is $-8$, so you would add $\left(\frac{-8}{2}\right)^2 = 16$ to each side of the equation.

Q: How do I rewrite the left-hand side of the equation as a perfect square?

A: To rewrite the left-hand side of the equation as a perfect square, you need to add the constant term that you added to each side of the equation to the left-hand side. For example, if you added $16$ to each side of the equation, you would rewrite the left-hand side as $(x - 4)^2$.

Q: How do I solve for $x$?

A: To solve for $x$, you need to take the square root of both sides of the equation. This will give you two possible solutions for $x$. You can then add or subtract the square root of the constant term from the left-hand side to get the two possible solutions.

Q: What are the two possible solutions for $x$?

A: The two possible solutions for $x$ are $x = 9$ and $x = -1$.

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

A: Yes, you can use the method of completing the square to solve any quadratic equation. However, you need to make sure that the equation is in the form $x^2 + bx = c$ and that you can rewrite the left-hand side as a perfect square.

Q: What are some examples of quadratic equations that can be solved using the method of completing the square?

A: Some examples of quadratic equations that can be solved using the method of completing the square include:

• $x^2 + 6x - 8 = 0$
• $x^2 - 2x - 15 = 0$
• $x^2 + 4x - 5 = 0$

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

A: You can check if the equation can be solved using the method of completing the square by looking at the coefficient of the $x$ term. If the coefficient is a multiple of $2$, then the equation can be solved using the method of completing the square.

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

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

• Not adding the constant term to both sides of the equation
• Not rewriting the left-hand side as a perfect square
• Not taking the square root of both sides of the equation

Q: How do I check my work when using the method of completing the square?

A: To check your work when using the method of completing the square, you can plug the solutions back into the original equation to make sure that they are true. You can also use a calculator to check your work.

Conclusion

In this article, we answered some common questions about the method of completing the square. We discussed how to rewrite the equation in the form $x^2 + bx = c$, how to complete the square, and how to solve for $x$. We also provided some examples of quadratic equations that can be solved using the method of completing the square and some common mistakes to avoid.