- Rearrange the equation to the standard form: \(x^2 + x + 2 = 0\).
- Calculate the discriminant: \(\Delta = 1^2 - 4(1)(2) = -7\).
- Since \(\Delta < 0\), the system has 0 solutions and never intersects.
- The system intersects in \(\boxed{0}\) place(s), and the statement that is true is \(\boxed{\text{the line and parabola do not intersect}}\).
Explanation
- Analyzing the System
The objective is to find the number of intersection points for the system defined by the equation \(-x^2 + x + 6 = 2x + 8\). An intersection point occurs where the graph of the function \(f(x) = -x^2 + x + 6\) (a parabola) and the graph of the function \(g(x) = 2x + 8\) (a line) meet. To find these points, we must solve for \(x\) in the equation \(-x^2 + x + 6 = 2x + 8\).
- Rearranging into Standard Form
To solve the equation, we move all terms to one side to set it into the standard quadratic form \(Ax^2 + Bx + C = 0\):
\[-x^2 + x - 2x + 6 - 8 = 0\]
\[-x^2 - x - 2 = 0\]
To simplify, we multiply the entire equation by \(-1\):
\[x^2 + x + 2 = 0\]
Now, we have a quadratic equation with coefficients \(A = 1\), \(B = 1\), and \(C = 2\).
- Calculating the Discriminant
The number of real solutions to a quadratic equation is determined by its discriminant, \(\Delta\), which is given by the formula \(\Delta = B^2 - 4AC\).
Substituting our values:
\[\Delta = (1)^2 - 4(1)(2)\]
\[\Delta = 1 - 8\]
\[\Delta = -7\]
Because the discriminant is negative (\(\Delta < 0\)), there are no real solutions to the equation. This indicates that the parabola and the line do not have any real intersection points.
- Conclusion
Since there are no real solutions to the equation \(-x^2 + x + 6 = 2x + 8\), the parabola and the line never intersect. Therefore, the system intersects in \(\boxed{0}\) place(s). The statement regarding the relationship between the parabola and the line is: \(\boxed{\text{the line and parabola do not intersect}}\).
Examples
Understanding the intersection of functions is crucial in fields like physics and engineering. For example, if you are modeling the trajectory of a ball as a parabola and a safety net as a line, calculating their intersection helps determine if the ball will hit the net or pass over it. If the equation describing their intersection has no real roots, it mathematically proves the ball will never strike the net.