Which of the following are roots of `4x^(2)-9x-100=0?`
(i) -4 (ii) `(3)/(4)` (iii) `(25)/(4)`

Answer

To determine which of the given options are roots of the quadratic equation \( 4x^2 - 9x - 100 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \).
For our equation:
- \( a = 4 \)
- \( b = -9 \)
- \( c = -100 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is calculated using the formula:
\[ D = b^2 - 4ac \]
Substituting the values:
\[ D = (-9)^2 - 4 \cdot 4 \cdot (-100) \]
\[ D = 81 + 1600 = 1681 \]

### Step 3: Use the quadratic formula to find the roots
The roots of the quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values of \( b \) and \( D \):
\[ x = \frac{-(-9) \pm \sqrt{1681}}{2 \cdot 4} \]
\[ x = \frac{9 \pm 41}{8} \]

### Step 4: Calculate the two possible values for \( x \)
1. For the positive case:
\[ x = \frac{9 + 41}{8} = \frac{50}{8} = \frac{25}{4} \]
2. For the negative case:
\[ x = \frac{9 - 41}{8} = \frac{-32}{8} = -4 \]

### Step 5: List the roots
The roots of the equation \( 4x^2 - 9x - 100 = 0 \) are:
- \( x = \frac{25}{4} \)
- \( x = -4 \)

### Step 6: Check the options
Now we check which of the given options are roots:
- (i) \( -4 \) (is a root)
- (ii) \( \frac{3}{4} \) (is not a root)
- (iii) \( \frac{25}{4} \) (is a root)

### Conclusion
The roots of the equation \( 4x^2 - 9x - 100 = 0 \) are \( -4 \) and \( \frac{25}{4} \). Thus, the correct options are (i) and (iii).

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