To determine the number of solutions to the equation [tex]\(-x^2 + x + 6 = 2x + 8\)[/tex], we'll start by simplifying and solving it step-by-step.
1. Start with the given equation:
[tex]\[
-x^2 + x + 6 = 2x + 8.
\][/tex]
2. Move all the terms to one side of the equation to set it to 0:
[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 have a quadratic equation:
[tex]\[
-x^2 - x - 2 = 0.
\][/tex]
5. Solve the quadratic equation using the quadratic formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
\][/tex]
For our equation, [tex]\(a = -1\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -2\)[/tex]. Substituting these values into the quadratic formula, we get:
[tex]\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-1)(-2)}}{2(-1)}.
\][/tex]
6. Simplify inside the square root:
[tex]\[
x = \frac{1 \pm \sqrt{1 - 8}}{-2}.
\][/tex]
7. Calculate the discriminant:
[tex]\[
1 - 8 = -7.
\][/tex]
Since the discriminant is negative, the square root of [tex]\(-7\)[/tex] is an imaginary number.
8. Express the solutions involving the imaginary unit [tex]\(i\)[/tex] (where [tex]\(i = \sqrt{-1}\)[/tex]):
[tex]\[
x = \frac{1 \pm \sqrt{-7}}{-2} = \frac{1 \pm \sqrt{7}i}{-2}.
\][/tex]
9. Separate into two solutions:
[tex]\[
x = -\frac{1}{2} - \frac{\sqrt{7}i}{2} \quad \text{and} \quad x = -\frac{1}{2} + \frac{\sqrt{7}i}{2}.
\][/tex]
Therefore, the solutions to the equation [tex]\(-x^2 + x + 6 = 2x + 8\)[/tex] 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]
10. Interpret the result:
These solutions are complex numbers (involving [tex]\(i\)[/tex]). Since the solutions are not real numbers, we can conclude that the equation has no real solutions.
To answer the multiple-choice question:
- The statement "The solution is [tex]\(x = 3\)[/tex]." is false.
- The statement "The solutions are [tex]\(x = -2\)[/tex] and [tex]\(x = 3\)[/tex]." is false.
- The statement "There are no solutions." is false in the general sense, but true if considering only real solutions.
- The statement "There are infinite solutions." is false.
The correct interpretation here is that there are no real solutions. Thus, among the options provided, the statement “There are no solutions” is closest to being true considering the context specified as only real solutions.
