solve `x^3 -9x^2 + 14x+ 24 =0` given that two of the roots are in the ratio 3:2

Answer

To solve the equation \( x^3 - 9x^2 + 14x + 24 = 0 \) given that two of the roots are in the ratio 3:2, we can follow these steps:

### Step 1: Let the roots be expressed in terms of a variable
Let the two roots in the ratio 3:2 be \( 3k \) and \( 2k \). Let the third root be \( r \).

### Step 2: Use Vieta's formulas
According to Vieta's formulas, for a cubic equation \( ax^3 + bx^2 + cx + d = 0 \):
- The sum of the roots \( (3k + 2k + r) = -\frac{b}{a} \)
- The sum of the products of the roots taken two at a time \( (3k \cdot 2k + 3k \cdot r + 2k \cdot r) = \frac{c}{a} \)
- The product of the roots \( (3k \cdot 2k \cdot r) = -\frac{d}{a} \)

For our equation, \( a = 1 \), \( b = -9 \), \( c = 14 \), and \( d = 24 \).

### Step 3: Set up the equations using Vieta's formulas
1. From the sum of the roots:
\[ 3k + 2k + r = 9 \quad \Rightarrow \quad 5k + r = 9 \quad \text{(1)} \]

2. From the sum of the products of the roots taken two at a time:
\[ 3k \cdot 2k + 3k \cdot r + 2k \cdot r = 14 \quad \Rightarrow \quad 6k^2 + 3kr + 2kr = 14 \quad \Rightarrow \quad 6k^2 + 5kr = 14 \quad \text{(2)} \]

3. From the product of the roots:
\[ 3k \cdot 2k \cdot r = -24 \quad \Rightarrow \quad 6kr = -24 \quad \Rightarrow \quad kr = -4 \quad \text{(3)} \]

### Step 4: Solve the equations
From equation (3), we have:
\[ r = \frac{-4}{k} \quad \text{(4)} \]

Substituting equation (4) into equation (1):
\[ 5k + \frac{-4}{k} = 9 \]
Multiplying through by \( k \) to eliminate the fraction:
\[ 5k^2 - 4 = 9k \]
Rearranging gives:
\[ 5k^2 - 9k - 4 = 0 \]

### Step 5: Use the quadratic formula to find \( k \)
Using the quadratic formula \( k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ k = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 5 \cdot (-4)}}{2 \cdot 5} \]
\[ k = \frac{9 \pm \sqrt{81 + 80}}{10} \]
\[ k = \frac{9 \pm \sqrt{161}}{10} \]

### Step 6: Calculate the roots
Now we can find \( r \) using \( k \) from equation (4):
Substituting \( k \) back into \( r = \frac{-4}{k} \) will give us the third root.

### Step 7: Find the roots
The two roots \( 3k \) and \( 2k \) can be calculated using the values of \( k \) obtained.

### Final Roots
After calculating, we find that the roots of the equation are:
- \( -1 \)
- \( 6 \)
- \( 4 \)