To solve the quadratic equation [tex]\(-x^2 + 9x + 7 = 0\)[/tex] and determine the number of real solutions, we can follow these steps:
1. Identify the coefficients: For the quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex], identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
- Here, [tex]\(a = -1\)[/tex], [tex]\(b = 9\)[/tex], and [tex]\(c = 7\)[/tex].
2. Calculate the discriminant: The discriminant of a quadratic equation is given by the formula [tex]\(D = b^2 - 4ac\)[/tex].
- Calculate the discriminant: [tex]\(D = 9^2 - 4(-1)(7)\)[/tex].
- This gives [tex]\(D = 81 + 28\)[/tex].
- Therefore, [tex]\(D = 109\)[/tex].
3. Analyze the discriminant:
- If [tex]\(D > 0\)[/tex], the equation has two distinct real solutions.
- If [tex]\(D = 0\)[/tex], the equation has exactly one real solution.
- If [tex]\(D < 0\)[/tex], the equation has no real solutions.
Since the discriminant [tex]\(D = 109\)[/tex] is greater than zero, this means the quadratic equation [tex]\(-x^2 + 9x + 7 = 0\)[/tex] has two distinct real solutions.
Therefore, the correct answer is:
c. two solutions
