Question

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

Answer 1

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

Answer 2

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

Answer 3

its not 1 or -9

Answer 4

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 5

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 6

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 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!