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 will substitute each option into the equation and check if it satisfies the equation.

### Step 1: Substitute \(x = -4\)

Substituting \(x = -4\) into the equation:

\[ 4(-4)^2 - 9(-4) - 100 = 0 \]

Calculating each term:

\[ 4(16) + 36 - 100 = 0 \]

\[ 64 + 36 - 100 = 0 \]

\[ 100 - 100 = 0 \]

Since the left-hand side equals 0, \(x = -4\) is a root of the equation.

### Step 2: Substitute \(x = \frac{3}{4}\)

Now, substituting \(x = \frac{3}{4}\):

\[ 4\left(\frac{3}{4}\right)^2 - 9\left(\frac{3}{4}\right) - 100 = 0 \]

Calculating each term:

\[ 4\left(\frac{9}{16}\right) - \frac{27}{4} - 100 = 0 \]

\[ \frac{36}{16} - \frac{27}{4} - 100 = 0 \]

Converting \(\frac{27}{4}\) to sixteenths:

\[ \frac{36}{16} - \frac{108}{16} - 100 = 0 \]

Combining the fractions:

\[ \frac{36 - 108}{16} - 100 = 0 \]

\[ \frac{-72}{16} - 100 = 0 \]

\[ -\frac{9}{2} - 100 \neq 0 \]

Since the left-hand side does not equal 0, \(x = \frac{3}{4}\) is not a root of the equation.

### Step 3: Substitute \(x = \frac{25}{4}\)

Now, substituting \(x = \frac{25}{4}\):

\[ 4\left(\frac{25}{4}\right)^2 - 9\left(\frac{25}{4}\right) - 100 = 0 \]

Calculating each term:

\[ 4\left(\frac{625}{16}\right) - \frac{225}{4} - 100 = 0 \]

\[ \frac{2500}{16} - \frac{900}{16} - 100 = 0 \]

Combining the fractions:

\[ \frac{2500 - 900}{16} - 100 = 0 \]

\[ \frac{1600}{16} - 100 = 0 \]

\[ 100 - 100 = 0 \]

Since the left-hand side equals 0, \(x = \frac{25}{4}\) is a root of the equation.

### Conclusion

The roots of the equation \(4x^2 - 9x - 100 = 0\) are \(x = -4\) and \(x = \frac{25}{4}\). Therefore, the correct options are:

- (i) -4
- (iii) \(\frac{25}{4}\)