Complete the square to identify the two values that solve the following quadratic equation: x^2 + 8x = 9 x = ?

Answer 1

To solve the quadratic equation \(x^2 + 8x = 9\) by completing the square, follow these steps:

1. Move the constant term to the other side of the equation: \[ x^2 + 8x - 9 = 0 \] or, equivalently, \[ x^2 + 8x = 9. \]

2. To complete the square, take the coefficient of \(x\) (which is 8), divide it by 2, and square it: \[ \left(\frac{8}{2}\right)^2 = 4^2 = 16. \]

3. Add and subtract this square (16) on the left side of the equation: \[ x^2 + 8x + 16 - 16 = 9, \] or rewriting this gives: \[ (x + 4)^2 - 16 = 9. \]

4. Now, rearrange the equation: \[ (x + 4)^2 = 9 + 16, \] which simplifies to: \[ (x + 4)^2 = 25. \]

5. Take the square root of both sides: \[ x + 4 = \pm 5. \]

6. Now, solve for \(x\) by isolating it in both cases: \[ x + 4 = 5 \quad \Rightarrow \quad x = 5 - 4 = 1, \] and \[ x + 4 = -5 \quad \Rightarrow \quad x = -5 - 4 = -9. \]

Thus, the two values that solve the quadratic equation \(x^2 + 8x = 9\) are: \[ x = 1 \quad \text{and} \quad x = -9. \]

Answer 2

Make sure to combine all constants using the inverse of addition or subtraction before determining the square root.

Answer 3

Let's go through the process of completing the square while paying closer attention to combining the constants properly before determining the square root.

Starting with the equation:

\[ x^2 + 8x = 9 \]

1. Move the constant to the other side:

   \[ x^2 + 8x - 9 = 0 \]

   or, in the original form:

   \[ x^2 + 8x - 9 = 0 \]

2. Complete the square:

   • Take the coefficient of \(x\), which is \(8\), divide by \(2\) to get \(4\), and then square it:

   \[ \left(\frac{8}{2}\right)^2 = 4^2 = 16 \]

3. Add and subtract \(16\) on the left side:

   \[ x^2 + 8x + 16 - 16 = 9 \]

   This can be rearranged to:

   \[ (x + 4)^2 - 16 = 9 \]

4. Combine constants on the right side:

   Rearrange the equation to isolate the perfect square:

   \[ (x + 4)^2 = 9 + 16 \]

   Simplifying the right side gives:

   \[ (x + 4)^2 = 25 \]

5. Take the square root of both sides:

   \[ x + 4 = \pm 5 \]

6. Solve for \(x\):

   • For \(x + 4 = 5\):

   \[ x = 5 - 4 = 1 \]

   • For \(x + 4 = -5\):

   \[ x = -5 - 4 = -9 \]

So the two values that solve the quadratic equation \(x^2 + 8x = 9\) are:

\[ \boxed{1} \quad \text{and} \quad \boxed{-9} \]

Thank you for your guidance, and I hope this process meets your expectations!

Answer 4

Complete the square to identify the two values that solve the following quadratic equation: x^2 + 8x = 9?

Answer 5

To solve the quadratic equation \(x^2 + 8x = 9\) by completing the square, follow these steps:

1. Rearrange the equation: Start by moving the constant term to the left side: \[ x^2 + 8x - 9 = 0 \] or keeping it in the equation form: \[ x^2 + 8x = 9. \]

2. Complete the square:

   • Take the coefficient of \(x\) (which is \(8\)), divide it by \(2\), and then square it: \[ \left(\frac{8}{2}\right)^2 = 4^2 = 16. \]

3. Add and subtract this square inside the equation: We rewrite the equation as: \[ x^2 + 8x + 16 - 16 = 9, \] which simplifies to: \[ (x + 4)^2 - 16 = 9. \]

4. Isolate the squared term: Now, move \(-16\) to the right side: \[ (x + 4)^2 = 9 + 16, \] which simplifies to: \[ (x + 4)^2 = 25. \]

5. Take the square root of both sides: \[ x + 4 = \pm 5. \]

6. Solve for \(x\):

   • For the positive case: \[ x + 4 = 5 \quad \Rightarrow \quad x = 5 - 4 = 1. \]
   • For the negative case: \[ x + 4 = -5 \quad \Rightarrow \quad x = -5 - 4 = -9. \]

Thus, the two values that solve the quadratic equation \(x^2 + 8x = 9\) are:

\[ \boxed{1} \quad \text{and} \quad \boxed{-9}. \]

Answer 6

its not 1 or -9

Answer 7

Apologies for any confusion in my previous responses. Let's go through the completion of the square step by step once again to ensure accuracy.

We start with the equation:

\[ x^2 + 8x = 9 \]

Step 1: Move the constant to the other side

Let's rearrange it:

\[ x^2 + 8x - 9 = 0 \]

Step 2: Complete the square

We need to complete the square for \(x^2 + 8x\).

1. Take the coefficient of \(x\) (which is \(8\)), divide it by \(2\) (giving \(4\)), and square it (giving \(16\)).

2. Rewrite the equation as:

   \[ x^2 + 8x + 16 - 16 = 9 \]

Reorganize it to group the completed square:

\[ (x + 4)^2 - 16 = 9 \]

Step 3: Simplify the equation

Now, add \(16\) to both sides:

\[ (x + 4)^2 = 25 \]

Step 4: Take the square root of both sides

Now we take the square root of both sides:

\[ x + 4 = \pm 5 \]

Step 5: Solve for \(x\)

1. If \(x + 4 = 5\): \[ x = 5 - 4 = 1 \]

2. If \(x + 4 = -5\): \[ x = -5 - 4 = -9 \]

Thus, the solutions obtained are indeed:

\[ x = 1 \quad \text{and} \quad x = -9. \]

Conclusion

If you have indicated that these are incorrect, please let me know the correct values you have or check the values in relation to the given equation \(x^2 + 8x = 9\) to ensure they match the calculation. Thank you for your patience!