To solve the equation [tex]\(-x^2 + x + 6 = 2x + 8\)[/tex] and determine which graph represents the system:
1. Write the original equation:
[tex]\[
-x^2 + x + 6 = 2x + 8
\][/tex]
2. Move all terms to one side to set the equation to zero:
[tex]\[
-x^2 + x + 6 - 2x - 8 = 0
\][/tex]
3. Combine like terms:
[tex]\[
-x^2 + x - 2x + 6 - 8 = 0
\][/tex]
Which simplifies to:
[tex]\[
-x^2 - x - 2 = 0
\][/tex]
4. Now, we solve for [tex]\(x\)[/tex] by recognizing this is a simple quadratic equation:
[tex]\[
-x^2 - x - 2 = 0
\][/tex]
5. Solve the quadratic equation:
We use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = -1\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -2\)[/tex].
First, we calculate the discriminant:
[tex]\[
b^2 - 4ac = (-1)^2 - 4(-1)(-2) = 1 - 8 = -7
\][/tex]
Since the discriminant is negative ([tex]\(-7\)[/tex]), the solutions will be complex (non-real).
6. Compute the roots:
[tex]\[
x = \frac{-(-1) \pm \sqrt{-7}}{2(-1)} = \frac{1 \pm \sqrt{7}i}{-2} = -\frac{1 \pm \sqrt{7}i}{2}
\][/tex]
Therefore, the solutions are:
[tex]\[
x = -\frac{1}{2} - \frac{\sqrt{7}i}{2} \quad \text{and} \quad x = -\frac{1}{2} + \frac{\sqrt{7}i}{2}
\][/tex]
7. Interpretation:
The fact that we have complex solutions means that the quadratic equation does not intersect the x-axis; instead, the parabolas of the equations [tex]\( -x^2 + x + 6 = y \)[/tex] and [tex]\( 2x + 8 = y \)[/tex] do not intersect on the real plane.
8. Graph Representation:
Since the solutions are complex, this also informs us that the graph of the system, consisting of the parabolas represented by the equations [tex]\( -x^2 + x + 6 \)[/tex] and [tex]\( 2x + 8 \)[/tex], will not intersect on the real coordinate plane. The graph representations will show two separate curves that do not touch at any point on the graph.
In summary, the graph of the system will represent two parabolas that do not intersect anywhere on the real plane.
