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Problem Details:

Given statements:

1. Statement I: If the roots of a quadratic equation are $2$ and $3$, then the equation is $x^2 - 5x + 6 = 0$.
2. Statement II: If the roots of $4x^2 + 3kx + 9 = 0$ are real and distinct, then $k \leq -4 \text{ or } k \geq 4$.

You need to determine the correctness of these statements.

Step-by-step Solution:

Statement I Verification:

1. Roots of a quadratic equation are given as $2$ and $3$.
2. A quadratic equation with roots $p$ and $q$ is given by: $(x - p)(x - q) = 0$ Substituting $p = 2$ and $q = 3$: $(x - 2)(x - 3) = 0 \quad \Rightarrow \quad x^2 - 5x + 6 = 0$ Hence, Statement I is correct.

Statement II Verification:

The quadratic equation is $4x^2 + 3kx + 9 = 0$. For the roots to be real and distinct, the discriminant ($\Delta$) must be positive: $\Delta = b^2 - 4ac$ Here, $a = 4$, $b = 3k$, and $c = 9$. Substituting: $\Delta = (3k)^2 - 4(4)(9)$ $\Delta = 9k^2 - 144$ For real and distinct roots: $\Delta > 0 \quad \Rightarrow \quad 9k^2 - 144 > 0$ Simplify: $9k^2 > 144 \quad \Rightarrow \quad k^2 > 16$ Taking the square root: $|k| > 4 \quad \Rightarrow \quad k < -4 \text{ or } k > 4$ Hence, Statement II is correct.

Final Answer:

Both statements are correct.
The correct option is:

• Option 1: Both Statement I and Statement II are correct

Do you want further clarification or details about any step? Let me know!

5 Related Questions to Expand Your Understanding:

1. How do you derive a quadratic equation from given roots?
2. What are the conditions for a quadratic equation to have real roots, distinct roots, or complex roots?
3. How can the discriminant of a quadratic equation be interpreted geometrically?
4. Can the roots of a quadratic equation be negative or imaginary? Provide examples.
5. How does changing the coefficient of $x^2$ in a quadratic equation affect the parabola?

1 Tip:

Always check the discriminant ($\Delta = b^2 - 4ac$) for determining the nature of roots in quadratic equations—this is a critical and time-saving step!