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.