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.