Find the set of values of k for which the curve y=(k+1)x^2-3x+(k+1) lies below the x-axis.

To find the set of values of \( k \) for which the curve \( y = (k+1)x^2 - 3x + (k+1) \) lies below the x-axis, we need to determine when the quadratic function is negative, i.e., \( y < 0 \) for all \( x \).

### Step 1: Analyze the quadratic function
The general form of a quadratic function is:
\[
y = ax^2 + bx + c
\]
In this case, \( a = k+1 \), \( b = -3 \), and \( c = k+1 \).

### Step 2: Conditions for the quadratic to be below the x-axis
For the quadratic function to lie completely below the x-axis, the following conditions must be satisfied:
1. The quadratic has no real roots (i.e., the discriminant must be negative).
2. The coefficient \( a = k+1 \) must be positive, ensuring that the parabola opens upwards.

### Step 3: Discriminant analysis
The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[
\Delta = b^2 - 4ac
\]
Substituting the values for \( a \), \( b \), and \( c \):
\[
\Delta = (-3)^2 - 4(k+1)(k+1)
\]
\[
\Delta = 9 - 4(k+1)^2
\]
For the quadratic to have no real roots, the discriminant must be negative:
\[
9 - 4(k+1)^2 < 0
\]
### Step 4: Solve the inequality
First, solve the inequality:
\[
9 < 4(k+1)^2
\]
Divide both sides by 4:
\[
\frac{9}{4} < (k+1)^2
\]
Taking the square root of both sides:
\[
\frac{3}{2} < |k+1|
\]
This implies two possible inequalities:
\[
k+1 > \frac{3}{2} \quad \text{or} \quad k+1 < -\frac{3}{2}
\]
Solving these:
\[
k > \frac{1}{2} \quad \text{or} \quad k < -\frac{5}{2}
\]

### Step 5: Ensure the parabola opens upwards
For the parabola to open upwards, the coefficient \( a = k+1 \) must be positive:
\[
k+1 > 0 \quad \Rightarrow \quad k > -1
\]

### Step 6: Combine the results
- From the discriminant, we have \( k > \frac{1}{2} \) or \( k < -\frac{5}{2} \).
- The parabola opens upwards if \( k > -1 \).

Thus, the final set of values for \( k \) for which the curve lies below the x-axis is:
\[
\boxed{k < -\frac{5}{2} \text{ or } k > \frac{1}{2}}
\]

Would you like further details or have any questions?

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Here are 5 related questions:
1. What is the significance of the discriminant in quadratic equations?
2. How would the solution change if we wanted the curve to lie above the x-axis?
3. What are the general conditions for a quadratic curve to intersect the x-axis?
4. How do you find the vertex of the parabola described by the quadratic equation?
5. What happens if \( k = -\frac{5}{2} \) or \( k = \frac{1}{2} \)?

**Tip:** Always check the sign of the leading coefficient \( a \) in a quadratic equation to understand the direction in which the parabola opens.

Find the set of values of k for which the curve y = (k+1)x^2 - 3x + (k+1) lies below the x-axis.

Discover how to find the set of values of k for which the curve y = (k+1)x^2 - 3x + (k+1) lies below the x-axis. This solution involves discriminant analysis and conditions for a quadratic function to be negative, making it ideal for students in Grades 10-12.

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