(x^2)-kx+2k-5=0 use the discriminant to solve for k

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?

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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.

Learn how to solve the quadratic equation (x^2)-kx+2k-5=0 by using the discriminant. This solution walks through key algebraic concepts, including the discriminant formula and the impact of different values of k on the roots. Suitable for students in Grades 9-12.

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