Can someone please tell me how to solve and graph this system of trinomial quadratic inequalities?
{y is greater than or equal to x^2+6x+12}
{y<-x^2-8x-12 }
To solve and graph the system of trinomial quadratic inequalities, you can follow these steps:
1. Solve each inequality separately to determine the boundaries of each solution region.
- For the first inequality, y ≥ x^2 + 6x + 12:
- To find the boundary, set y = x^2 + 6x + 12 and solve for x.
Let's complete the square to convert the quadratic equation into vertex form:
x^2 + 6x + 12 = y
x^2 + 6x + 9 = y - 12 + 9
(x + 3)^2 = y - 3
- The parabola opens upward, and the vertex is at (-3, 3).
- Since y is greater than or equal to the expression, the solution region will be above or on the parabola.
- For the second inequality, y < -x^2 - 8x - 12:
- To find the boundary, set y = -x^2 - 8x - 12 and solve for x.
Let's complete the square again:
x^2 + 8x + 12 = -y
x^2 + 8x + 4^2 = -y + 12 + 4^2
(x + 4)^2 = -y + 28
- The parabola opens downward, and the vertex is at (-4, 28).
- Since y is less than the expression, the solution region will be below the parabola.
2. Graph the boundary lines.
- Plot the vertex of each parabola on the coordinate plane.
- Draw the parabola using the vertex as the symmetric point.
3. Determine the solution region.
- For the first inequality, y ≥ x^2 + 6x + 12, shade the region above or on the parabola.
- For the second inequality, y < -x^2 - 8x - 12, shade the region below the parabola.
4. Determine the overlapping region.
- The solution to the system of inequalities is the overlapping region between the shaded areas.
Note: You can use graphing software or online graphing tools to visualize the graphs and find the solution region.