The roots of the equation `3a^(2)x^(2)+8abx+4b^(2)=0`,where `ane0` are

Options

A. `(3b)/(2a),(b)/(3a)`
B. `(-2b)/(3a),-(2b)/(a)`
C. `(b)/(3a),(2b)/(3a)`
D. `(2a)/(b),(4b)/(a)`

Answer

To find the roots of the quadratic equation \(3a^2x^2 + 8abx + 4b^2 = 0\), we will use the quadratic formula, which is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 1: Identify the coefficients
In the equation \(3a^2x^2 + 8abx + 4b^2 = 0\), we identify the coefficients as follows:
- \(A = 3a^2\)
- \(B = 8ab\)
- \(C = 4b^2\)

### Step 2: Substitute the coefficients into the quadratic formula
Now we substitute these coefficients into the quadratic formula:

\[ x = \frac{-8ab \pm \sqrt{(8ab)^2 - 4 \cdot 3a^2 \cdot 4b^2}}{2 \cdot 3a^2} \]

### Step 3: Calculate the discriminant
Next, we calculate the discriminant \(D = B^2 - 4AC\):

\[ D = (8ab)^2 - 4 \cdot 3a^2 \cdot 4b^2 \]
\[ D = 64a^2b^2 - 48a^2b^2 \]
\[ D = 16a^2b^2 \]

### Step 4: Substitute the discriminant back into the formula
Now we substitute \(D\) back into the quadratic formula:

\[ x = \frac{-8ab \pm \sqrt{16a^2b^2}}{6a^2} \]

### Step 5: Simplify the square root
The square root of \(16a^2b^2\) is \(4ab\):

\[ x = \frac{-8ab \pm 4ab}{6a^2} \]

### Step 6: Split into two cases
This gives us two potential roots:

1. When using the positive sign:
\[ x_1 = \frac{-8ab + 4ab}{6a^2} = \frac{-4ab}{6a^2} = \frac{-2b}{3a} \]

2. When using the negative sign:
\[ x_2 = \frac{-8ab - 4ab}{6a^2} = \frac{-12ab}{6a^2} = \frac{-2b}{a} \]

### Step 7: Final roots
Thus, the roots of the equation are:

\[ x_1 = \frac{-2b}{3a} \quad \text{and} \quad x_2 = \frac{-2b}{a} \]