To solve the quadratic inequality $y < x^2 + 2x - 4$, we follow these steps:

Step 1: Understand the inequality

The given inequality is $y < x^2 + 2x - 4$, which describes the region below the parabola defined by $y = x^2 + 2x - 4$.

Step 2: Analyze the quadratic expression

The quadratic expression $x^2 + 2x - 4$ represents a parabola that opens upwards since the coefficient of $x^2$ is positive. The vertex of this parabola occurs at the point where the derivative $2x + 2 = 0$, giving $x = -1$.

Substitute $x = -1$ into the equation to find the vertex: $y = (-1)^2 + 2(-1) - 4 = 1 - 2 - 4 = -5$ So, the vertex of the parabola is at $(-1, -5)$.

Step 3: Graph the inequality

• The equation $y = x^2 + 2x - 4$ represents a parabola with its vertex at $(-1, -5)$, and it opens upward.
• The inequality $y < x^2 + 2x - 4$ represents all the points below this parabola.

Step 4: Interpret the solution

• In terms of $x$, there is no specific range for $x$ values where the inequality holds. The solution is a region in the $xy$-plane where the $y$-values are less than the $y$-values on the parabola.

Final Answer:

The solution to the inequality $y < x^2 + 2x - 4$ is the set of all points $(x, y)$ in the plane where the $y$-coordinate is less than the value of $x^2 + 2x - 4$ for a given $x$-coordinate.

Would you like more details or have any questions?

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Here are 5 related questions to expand on this topic:

1. How do you solve quadratic inequalities in one variable?
2. What is the significance of the vertex in a parabola?
3. How do we determine the direction of a parabola's opening?
4. How do you graph quadratic inequalities?
5. What is the difference between solving quadratic inequalities and equations?

Tip: Always remember that the solution to a quadratic inequality involves regions in the coordinate plane, not just specific points!