What is the value of m in the quadratic equaiton ` x^(2) + mx + 24 ` = 0 if one of its roots is `(3)/(2)`

Options

A. `-(45)/(2)`
B. 16
C. `-(21)/(2)`
D. `-(35)/(2)`

Answer

To find the value of \( m \) in the quadratic equation \( x^2 + mx + 24 = 0 \) given that one of its roots is \( \frac{3}{2} \), we can follow these steps:

### Step 1: Substitute the root into the equation
Since \( \frac{3}{2} \) is a root of the equation, we can substitute \( x = \frac{3}{2} \) into the equation:

\[ \left(\frac{3}{2}\right)^2 + m\left(\frac{3}{2}\right) + 24 = 0 \]

### Step 2: Calculate \( \left(\frac{3}{2}\right)^2 \)
Calculating \( \left(\frac{3}{2}\right)^2 \):

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

### Step 3: Substitute back into the equation
Now substitute \( \frac{9}{4} \) back into the equation:

\[ \frac{9}{4} + m\left(\frac{3}{2}\right) + 24 = 0 \]

### Step 4: Convert 24 to a fraction
To add \( 24 \) to \( \frac{9}{4} \), we convert \( 24 \) into a fraction with a denominator of \( 4 \):

\[ 24 = \frac{24 \times 4}{4} = \frac{96}{4} \]

### Step 5: Combine the fractions
Now combine the fractions:

\[ \frac{9}{4} + \frac{96}{4} + m\left(\frac{3}{2}\right) = 0 \]

This simplifies to:

\[ \frac{105}{4} + m\left(\frac{3}{2}\right) = 0 \]

### Step 6: Isolate \( m \)
Now, isolate \( m\left(\frac{3}{2}\right) \):

\[ m\left(\frac{3}{2}\right) = -\frac{105}{4} \]

### Step 7: Solve for \( m \)
To solve for \( m \), multiply both sides by \( \frac{2}{3} \):

\[ m = -\frac{105}{4} \cdot \frac{2}{3} \]

Calculating this gives:

\[ m = -\frac{210}{12} = -\frac{35}{2} \]

### Final Answer
Thus, the value of \( m \) is:

\[ \boxed{-\frac{35}{2}} \]