To solve for $k$ using the discriminant, we will follow these steps:

Step 1: Recall the quadratic formula and discriminant

For a general quadratic equation $ax^2 + bx + c = 0$, the discriminant is given by:

$\Delta = b^2 - 4ac$

The discriminant determines the nature of the roots:

• If $\Delta > 0$, there are two real and distinct solutions.
• If $\Delta = 0$, there is one real and repeated solution.
• If $\Delta < 0$, there are no real solutions (the roots are complex).

Step 2: Identify coefficients in the equation

Your equation is $x^2 - kx + (2k - 5) = 0$. Here:

• $a = 1$ (coefficient of $x^2$),
• $b = -k$ (coefficient of $x$),
• $c = 2k - 5$ (constant term).

Step 3: Apply the discriminant formula

The discriminant is:

$\Delta = b^2 - 4ac = (-k)^2 - 4(1)(2k - 5)$

Simplify this expression:

$\Delta = k^2 - 4(2k - 5)$

Distribute the 4:

$\Delta = k^2 - 8k + 20$

Step 4: Solve for $k$ based on the desired conditions

Depending on what you want for the roots of the equation, you set the discriminant equal to:

• $\Delta = 0$ for one real solution,
• $\Delta > 0$ for two real solutions,
• $\Delta < 0$ for no real solutions.

Would you like to solve for $k$ with a specific condition, such as ensuring real or complex solutions?

---

Here are 5 related questions:

1. What are the conditions for a quadratic equation to have complex roots?
2. How can we interpret the role of the discriminant in graphing a quadratic function?
3. Can you solve the equation for $k$ under the condition $\Delta = 0$?
4. How does the value of $k$ affect the symmetry of the quadratic function's graph?
5. What happens to the nature of the roots as $k$ increases or decreases?

Tip: The discriminant provides key insight into the behavior of quadratic functions, especially for understanding the types of roots without solving the full equation.