To determine how many real and imaginary solutions the equation [tex]\(x^2 - 3x = -2x - 7\)[/tex] has, we need to solve it step-by-step.
1. First, bring all terms to one side to have the equation in the standard quadratic form:
[tex]\[ x^2 - 3x + 2x + 7 = 0 \][/tex]
2. Simplify the equation:
[tex]\[ x^2 - x + 7 = 0 \][/tex]
3. Now, we solve for [tex]\(x\)[/tex] using 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]\(c = 7\)[/tex].
4. Calculate the discriminant [tex]\(\Delta = b^2 - 4ac\)[/tex]:
[tex]\[ \Delta = (-1)^2 - 4(1)(7) \][/tex]
[tex]\[ \Delta = 1 - 28 \][/tex]
[tex]\[ \Delta = -27 \][/tex]
5. Analyzing the discriminant:
- If [tex]\(\Delta > 0\)[/tex], there are 2 distinct real solutions.
- If [tex]\(\Delta = 0\)[/tex], there is 1 real solution.
- If [tex]\(\Delta < 0\)[/tex], there are no real solutions, but there are 2 complex (imaginary) solutions.
Since [tex]\(\Delta = -27\)[/tex] is less than 0, there are no real solutions. Instead, we have 2 complex solutions.
Therefore, the equation [tex]\(x^2 - 3x = -2x - 7\)[/tex] has:
- 0 real solutions
- 2 imaginary solutions
So the correct choice is:
No real solutions; 2 imaginary solutions.
