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The given quadratic equation is:

\[
3x^2 - 4x + 6 = 0
\]

We are asked to find the following based on the fact that \(\alpha\) and \(\beta\) are the roots of this equation:

- (a) \(\alpha + \beta\)
- (b) \(\alpha \beta\)
- (c) \(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\)

### Step 1: Using Vieta's Formulas
For a quadratic equation of the form \(ax^2 + bx + c = 0\), Vieta's formulas provide the following relationships:
1. Sum of the roots: \(\alpha + \beta = -\frac{b}{a}\)
2. Product of the roots: \(\alpha \beta = \frac{c}{a}\)

Here, \(a = 3\), \(b = -4\), and \(c = 6\).

#### (a) Find \(\alpha + \beta\):
\[
\alpha + \beta = -\frac{-4}{3} = \frac{4}{3}
\]

#### (b) Find \(\alpha \beta\):
\[
\alpha \beta = \frac{6}{3} = 2
\]

### Step 2: Finding \(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\)
We use the identity:
\[
\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\alpha^2 + \beta^2}{\alpha^2 \beta^2}
\]
Now, recall that:
\[
\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta
\]
Substituting values:
\[
\alpha^2 + \beta^2 = \left(\frac{4}{3}\right)^2 - 2(2) = \frac{16}{9} - 4 = \frac{16}{9} - \frac{36}{9} = \frac{-20}{9}
\]
Also, \(\alpha^2 \beta^2 = (\alpha \beta)^2 = 2^2 = 4\).

Thus:
\[
\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\frac{-20}{9}}{4} = \frac{-20}{36} = \frac{-5}{9}
\]

### Final Answers:
- (a) \(\alpha + \beta = \frac{4}{3}\)
- (b) \(\alpha \beta = 2\)
- (c) \(\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{-5}{9}\)

Would you like more details on any of these steps or have further questions?

Here are some related questions you might consider:
1. How can Vieta's formulas be derived?
2. What is the relationship between the discriminant and the nature of the roots?
3. How can we solve a quadratic equation using the quadratic formula?
4. What is the geometric meaning of the sum and product of the roots?
5. How do we calculate higher powers of \(\alpha\) and \(\beta\) (e.g., \(\alpha^3 + \beta^3\))?

**Tip**: When dealing with roots of a quadratic equation, Vieta's relations are extremely helpful in quickly determining sums and products without directly solving the equation!

If α, β are roots of 3x^2 - 4x + 6 = 0, then find (a) α + β, (b) αβ, (c) 1/α^2 + 1/β^2

Learn how to find the sum and product of roots, and calculate 1/α^2 + 1/β^2 for the quadratic equation 3x^2 - 4x + 6 = 0. This solution uses Vieta's formulas and is suitable for students in Grades 9-11.

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