Which term must be added and subtracted to solve the quadratic equation `3x^2-5x+2=0` by the method of completing the square?

Options

A. 25
B. `25/4`
C. `25/9`
D. `25/35`

Answer

To solve the quadratic equation \(3x^2 - 5x + 2 = 0\) by the method of completing the square, we will follow these steps:

### Step 1: Rearrange the Equation
First, we want to isolate the quadratic and linear terms. We can do this by factoring out the coefficient of \(x^2\) from the left side of the equation.

\[ 3x^2 - 5x + 2 = 0 \]

Dividing the entire equation by 3 gives us:

\[ x^2 - \frac{5}{3}x + \frac{2}{3} = 0 \]

### Step 2: Move the Constant to the Right Side
Next, we will move the constant term to the right side of the equation:

\[ x^2 - \frac{5}{3}x = -\frac{2}{3} \]

### Step 3: Complete the Square
Now, we need to complete the square for the left-hand side. To do this, we take the coefficient of \(x\), which is \(-\frac{5}{3}\), halve it, and then square it.

Halving \(-\frac{5}{3}\):

\[ -\frac{5}{6} \]

Now squaring it:

\[ \left(-\frac{5}{6}\right)^2 = \frac{25}{36} \]

### Step 4: Add and Subtract the Square
We add and subtract \(\frac{25}{36}\) to the left side:

\[ x^2 - \frac{5}{3}x + \frac{25}{36} - \frac{25}{36} = -\frac{2}{3} \]

This simplifies to:

\[ \left(x - \frac{5}{6}\right)^2 - \frac{25}{36} = -\frac{2}{3} \]

### Step 5: Simplify the Equation
Now we need to combine the constants on the right side. We convert \(-\frac{2}{3}\) to have a denominator of 36:

\[ -\frac{2}{3} = -\frac{24}{36} \]

So we have:

\[ \left(x - \frac{5}{6}\right)^2 - \frac{25}{36} = -\frac{24}{36} \]

This gives us:

\[ \left(x - \frac{5}{6}\right)^2 - \frac{25}{36} + \frac{24}{36} = 0 \]

### Step 6: Final Form
Thus, we can rewrite the equation as:

\[ \left(x - \frac{5}{6}\right)^2 - \frac{1}{36} = 0 \]

### Conclusion
The term that must be added and subtracted to solve the quadratic equation \(3x^2 - 5x + 2 = 0\) by the method of completing the square is \(\frac{25}{36}\).

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