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### Question 6:
You are asked to analyze the graph of the quadratic function \( y = ax^2 + bx + c \), and determine whether each expression in the table is positive, negative, or zero based on the graph.

- \( b^2 - 4ac \): This is the **discriminant** of the quadratic equation, and it helps determine the nature of the roots of the equation.
  - If \( b^2 - 4ac > 0 \), the equation has **two distinct real roots**.
  - If \( b^2 - 4ac = 0 \), the equation has **one real root** (repeated or double root).
  - If \( b^2 - 4ac < 0 \), the equation has **no real roots** (complex roots).
  
- \( b \): This refers to the linear coefficient of the quadratic equation.

From the graph, we see that the parabola touches the x-axis once, which means the quadratic has **equal roots**. Therefore:
- \( b^2 - 4ac = 0 \) (discriminant is zero because the equation has equal roots).
- You need to determine the sign of \( b \) based on the orientation of the graph and the values of the vertex.

#### Fill out the table based on the information from the graph.

### Question 7:
You need to find the exact values of \( k \) for which the equation \( kx^2 + (k+1)x + 1 = 0 \) has **equal roots**. 

For the quadratic equation \( ax^2 + bx + c = 0 \) to have equal roots, its **discriminant** must be equal to 0:
\[
b^2 - 4ac = 0
\]
In your equation \( kx^2 + (k+1)x + 1 = 0 \), the coefficients are:
- \( a = k \)
- \( b = k+1 \)
- \( c = 1 \)

Now, apply the discriminant condition:
\[
(k+1)^2 - 4(k)(1) = 0
\]

Expanding and solving:
\[
(k+1)^2 - 4k = 0
\]
\[
k^2 + 2k + 1 - 4k = 0
\]
\[
k^2 - 2k + 1 = 0
\]
This simplifies to:
\[
(k - 1)^2 = 0
\]
Thus, \( k = 1 \).

### Answer:
The exact value of \( k \) for which the equation has equal roots is \( k = 1 \).

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Would you like further details or clarifications? Here are 5 related questions to expand your understanding:

1. What is the geometric interpretation of the discriminant in quadratic equations?
2. How does the discriminant change the number of real solutions to a quadratic equation?
3. Can the discriminant be applied to other types of polynomials?
4. How does changing the value of \( a \) in a quadratic equation affect the graph of the parabola?
5. What is the significance of a "double root" in quadratic equations?

**Tip**: When solving quadratic equations, always check the discriminant first to determine the nature of the roots.

Find the exact values of k for which the equation kx^2 + (k+1)x + 1 = 0 has equal roots.

This guide helps you find the exact value of k for which the quadratic equation kx^2 + (k+1)x + 1 = 0 has equal roots. It covers the discriminant theorem and demonstrates the steps to solve for k. Suitable for Grades 9-11 students learning algebra and quadratic equations.

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