if the roots of 3x^2+kx+6=0 are in ratio 2:3 product of all values of k =

To solve this, let's denote the roots of the quadratic equation \( 3x^2 + kx + 6 = 0 \) as \( \alpha \) and \( \beta \). According to the problem, the roots are in the ratio \( 2:3 \). This means we can let:

\[
\alpha = 2p \quad \text{and} \quad \beta = 3p
\]

### Step 1: Sum of the Roots
By Vieta's formulas, the sum of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[
\alpha + \beta = -\frac{b}{a}
\]

For the equation \( 3x^2 + kx + 6 = 0 \), we have \( a = 3 \), \( b = k \), and \( c = 6 \). Hence,

\[
\alpha + \beta = -\frac{k}{3}
\]

Substituting \( \alpha = 2p \) and \( \beta = 3p \):

\[
2p + 3p = -\frac{k}{3}
\]

\[
5p = -\frac{k}{3}
\]

Multiplying both sides by 3:

\[
15p = -k \quad \text{(Equation 1)}
\]

### Step 2: Product of the Roots
The product of the roots of a quadratic equation is given by:

\[
\alpha \beta = \frac{c}{a}
\]

For the equation \( 3x^2 + kx + 6 = 0 \), we have:

\[
\alpha \beta = \frac{6}{3} = 2
\]

Substituting \( \alpha = 2p \) and \( \beta = 3p \):

\[
(2p)(3p) = 2
\]

\[
6p^2 = 2
\]

Solving for \( p^2 \):

\[
p^2 = \frac{2}{6} = \frac{1}{3}
\]

Taking the square root of both sides:

\[
p = \frac{1}{\sqrt{3}}
\]

### Step 3: Finding \( k \)
Now, substitute \( p = \frac{1}{\sqrt{3}} \) into Equation 1:

\[
15p = -k
\]

\[
15 \times \frac{1}{\sqrt{3}} = -k
\]

\[
k = -15 \times \frac{1}{\sqrt{3}} = -\frac{15\sqrt{3}}{3} = -5\sqrt{3}
\]

Thus, the value of \( k \) is \( -5\sqrt{3} \).

### Step 4: Product of All Possible Values of \( k \)
There is only one value of \( k \), which is \( -5\sqrt{3} \). Hence, the product of all values of \( k \) is:

\[
\boxed{-5\sqrt{3}}
\]

Let me know if you'd like further details or clarifications!

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Here are 5 related questions to consider:
1. How do you determine the sum of the roots using Vieta's formulas?
2. How does the ratio of roots affect the setup of the problem?
3. What is the significance of solving for \( p \) in quadratic root problems?
4. How would the problem change if the ratio of the roots were different?
5. Can Vieta's formulas be applied to other types of polynomials?

**Tip:** Vieta's formulas are a powerful tool for connecting the coefficients of a polynomial to its roots without having to solve for them explicitly.

Solve for k in the quadratic equation 3x^2 + kx + 6 = 0 where the roots are in the ratio 2:3. Learn to use Vieta's formulas to determine the sum and product of the roots, and explore algebraic methods for solving quadratic equations.

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