If sum of the two roots of the equation `4x^2+16x^2-9x-36=0` is the first whole number, then the third root of this equation is:

Options

A. `-3//4`
B. `3//2`
C. `-3//2`
D. `-4`

Answer

To solve the equation \(4x^2 + 16x^2 - 9x - 36 = 0\) and find the third root given that the sum of the two roots is the first whole number, we will follow these steps:

### Step 1: Combine like terms
First, we combine the like terms in the equation:
\[ 4x^2 + 16x^2 = 20x^2 \]
So the equation simplifies to:
\[ 20x^2 - 9x - 36 = 0 \]

### Step 2: Apply the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 20\), \(b = -9\), and \(c = -36\).

### Step 3: Calculate the discriminant
First, we calculate the discriminant \(D\):
\[ D = b^2 - 4ac = (-9)^2 - 4 \cdot 20 \cdot (-36) \]
\[ D = 81 + 2880 = 2961 \]

### Step 4: Calculate the roots
Now we substitute \(D\) back into the quadratic formula:
\[ x = \frac{-(-9) \pm \sqrt{2961}}{2 \cdot 20} \]
\[ x = \frac{9 \pm \sqrt{2961}}{40} \]

### Step 5: Find the sum of the roots
The sum of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is given by:
\[ \text{Sum of roots} = -\frac{b}{a} = -\frac{-9}{20} = \frac{9}{20} \]
Since the problem states that the sum of the two roots is the first whole number, we need to check if this condition is satisfied.

### Step 6: Identify the third root
In a cubic equation, if we have three roots \(r_1\), \(r_2\), and \(r_3\), and we know the sum of two roots \(r_1 + r_2\), we can find the third root \(r_3\) using:
\[ r_3 = \text{Sum of roots} - (r_1 + r_2) \]
Given that the sum of the two roots is 1 (the first whole number), we can write:
\[ r_3 = \frac{9}{20} - 1 = \frac{9}{20} - \frac{20}{20} = -\frac{11}{20} \]

### Conclusion
Thus, the third root of the equation is:
\[ \text{Third root} = -\frac{11}{20} \]