Which constant must be added and subtracted to solve the quadratic equation `9x^2+ 3/4x - sqrt2 = 0` by the method of completing the square?

Options

A. `1/8`
B. `1/64`
C. `1/4`
D. `9/64`

Explanation

To solve the quadratic equation \(9x^2 + \frac{3}{4}x - \sqrt{2} = 0\) by the method of completing the square, we need to determine the constant that must be added and subtracted. Here are the steps to find that constant:

### Step 1: Rewrite the equation
We start with the equation:
\[ 9x^2 + \frac{3}{4}x - \sqrt{2} = 0 \]
To make completing the square easier, we can first isolate the quadratic and linear terms:
\[ 9x^2 + \frac{3}{4}x = \sqrt{2} \]

### Step 2: Factor out the coefficient of \(x^2\)
Next, we factor out the coefficient of \(x^2\) from the left side:
\[ 9\left(x^2 + \frac{1}{12}x\right) = \sqrt{2} \]

### Step 3: Identify the coefficient of \(x\)
Now, we need to complete the square for the expression inside the parentheses. The coefficient of \(x\) is \(\frac{1}{12}\).

### Step 4: Calculate half of the coefficient and square it
To complete the square, we take half of the coefficient of \(x\) (which is \(\frac{1}{12}\)), square it, and find the constant to be added and subtracted:
\[ \text{Half of } \frac{1}{12} = \frac{1}{24} \]
Now, squaring it gives:
\[ \left(\frac{1}{24}\right)^2 = \frac{1}{576} \]

### Step 5: Add and subtract the constant inside the parentheses
We add and subtract this constant inside the parentheses:
\[ 9\left(x^2 + \frac{1}{12}x + \frac{1}{576} - \frac{1}{576}\right) = \sqrt{2} \]
This simplifies to:
\[ 9\left(\left(x + \frac{1}{24}\right)^2 - \frac{1}{576}\right) = \sqrt{2} \]

### Step 6: Distribute the 9
Distributing the 9 gives:
\[ 9\left(x + \frac{1}{24}\right)^2 - \frac{9}{576} = \sqrt{2} \]

### Step 7: Simplify the constant
We can simplify \(\frac{9}{576}\):
\[ \frac{9}{576} = \frac{1}{64} \]
Thus, we rewrite the equation as:
\[ 9\left(x + \frac{1}{24}\right)^2 - \frac{1}{64} = \sqrt{2} \]

### Conclusion
The constant that must be added and subtracted to solve the quadratic equation by completing the square is:
\[ \frac{1}{576} \]