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The problem provided in the image asks about a function \( f(x) = 2x^2 - 3x + k \) and involves finding specific values related to this function. Here's a step-by-step breakdown:

### Problem Statement:
1. **Given**: \( f(x) = 2x^2 - 3x + k \), where \( k \) is a constant.
2. **Tasks**:
   - (a) Find the condition on \( k \) such that the equation \( f(x) = 0 \) has equal roots.
   - (b) When \( k = 7 \), find the roots of \( f(x) = 0 \), and determine the sum and product of the roots.

### Solution:

#### Part (a): Condition for Equal Roots
For the quadratic equation \( f(x) = 0 \), which is \( 2x^2 - 3x + k = 0 \), the condition for the roots to be equal is that the discriminant should be zero.

The discriminant \( \Delta \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[
\Delta = b^2 - 4ac
\]

For \( f(x) = 2x^2 - 3x + k \), we have:
- \( a = 2 \)
- \( b = -3 \)
- \( c = k \)

The discriminant is:

\[
\Delta = (-3)^2 - 4(2)(k) = 9 - 8k
\]

For the roots to be equal, \( \Delta \) must be zero:

\[
9 - 8k = 0
\]

Solving for \( k \):

\[
8k = 9 \quad \Rightarrow \quad k = \frac{9}{8}
\]

So, the condition for the roots to be equal is \( k = \frac{9}{8} \).

#### Part (b): Roots of the Equation when \( k = 7 \)
Now, we substitute \( k = 7 \) into the equation:

\[
f(x) = 2x^2 - 3x + 7 = 0
\]

The roots of a quadratic equation \( ax^2 + bx + c = 0 \) are given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting the values:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(7)}}{2(2)} = \frac{3 \pm \sqrt{9 - 56}}{4} = \frac{3 \pm \sqrt{-47}}{4}
\]

Since the discriminant is negative, the roots are complex:

\[
x = \frac{3 \pm i\sqrt{47}}{4}
\]

#### Sum and Product of the Roots:
- **Sum of the roots**: The sum of the roots for any quadratic equation \( ax^2 + bx + c = 0 \) is given by \( -\frac{b}{a} \):

\[
\text{Sum} = -\frac{-3}{2} = \frac{3}{2}
\]

- **Product of the roots**: The product of the roots is given by \( \frac{c}{a} \):

\[
\text{Product} = \frac{7}{2}
\]

### Final Answers:
1. The condition for the roots of \( f(x) = 0 \) to be equal is \( k = \frac{9}{8} \).
2. For \( k = 7 \), the roots of \( f(x) = 0 \) are complex, given by \( \frac{3 \pm i\sqrt{47}}{4} \). The sum of the roots is \( \frac{3}{2} \) and the product of the roots is \( \frac{7}{2} \).

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Would you like more details or have any other questions? Here are some related questions to explore:

1. What is the geometric interpretation of equal roots in a quadratic equation?
2. How do you determine the nature of roots using the discriminant?
3. What are the implications of complex roots in a real-world scenario?
4. How does the value of \( k \) affect the graph of the quadratic function \( f(x) = 2x^2 - 3x + k \)?
5. Can you derive the quadratic formula from completing the square?

**Tip:** The discriminant is a powerful tool to quickly assess the nature of the roots of a quadratic equation without actually solving it.

Explore how to solve quadratic equations with specific conditions, including finding equal roots and dealing with complex roots. This problem involves using the discriminant formula and understanding the nature of roots in quadratic equations. Suitable for Grades 10-12 students.

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