We are given the quadratic equation $2x^2 - 8x - k = 0$ with one of its roots being $\frac{5}{2}$. Our goal is to find the other root and the value of $k$.

Step 1: Use the sum of roots formula

For a quadratic equation of the form $ax^2 + bx + c = 0$, the sum of the roots is given by: $\text{Sum of roots} = -\frac{b}{a}$ In this case, $a = 2$ and $b = -8$. Therefore, the sum of the roots is: $\text{Sum of roots} = -\frac{-8}{2} = 4$

Let the other root be denoted by $r$. Since one root is $\frac{5}{2}$, the sum of the roots is: $\frac{5}{2} + r = 4$ Solving for $r$: $r = 4 - \frac{5}{2} = \frac{8}{2} - \frac{5}{2} = \frac{3}{2}$ Thus, the other root is $\frac{3}{2}$.

Step 2: Use the product of roots formula

The product of the roots for a quadratic equation is given by: $\text{Product of roots} = \frac{c}{a}$ Here, the product of the roots $\frac{5}{2}$ and $\frac{3}{2}$ is: $\frac{5}{2} \times \frac{3}{2} = \frac{15}{4}$ Since the product of the roots is equal to $\frac{c}{a}$, and $c = -k$ and $a = 2$, we have: $\frac{-k}{2} = \frac{15}{4}$ Multiplying both sides by 2: $-k = \frac{15}{2}$ Thus, $k = -\frac{15}{2}$.

Final Answer:

• The other root is $\frac{3}{2}$.
• The value of $k$ is $-\frac{15}{2}$.

Do you want further details or have any questions?

Here are 5 related questions:

1. How do you derive the sum and product of roots formula?
2. What happens if the roots of a quadratic equation are complex?
3. How do you factor a quadratic equation given its roots?
4. Can we solve this using the quadratic formula instead?
5. How does the discriminant of this quadratic equation compare to others?

Tip: Always check if the sum and product of the roots match the coefficients of the equation to verify correctness.