To determine the number of real solutions for the quadratic equation [tex]\(x^2 - 9 = 0\)[/tex], we follow these steps:
1. Rewrite the equation in the standard form of a quadratic equation: The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. For the given equation:
[tex]\[x^2 - 9 = 0\][/tex]
we can identify the coefficients as [tex]\(a = 1\)[/tex], [tex]\(b = 0\)[/tex], and [tex]\(c = -9\)[/tex].
2. Calculate the discriminant: The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by the formula [tex]\(\Delta = b^2 - 4ac\)[/tex]. Substituting the known values:
[tex]\[\Delta = b^2 - 4ac = (0)^2 - 4(1)(-9) = 0 - (-36) = 36\][/tex]
3. Determine the number of real solutions based on the discriminant:
- If the discriminant, [tex]\(\Delta\)[/tex], is greater than 0, the equation has 2 distinct real solutions.
- If the discriminant, [tex]\(\Delta\)[/tex], is equal to 0, the equation has exactly 1 real solution.
- If the discriminant, [tex]\(\Delta\)[/tex], is less than 0, the equation has no real solutions.
4. Analyze the calculated discriminant: In this case, the discriminant [tex]\(\Delta\)[/tex] is 36, which is greater than 0.
Therefore, the quadratic equation [tex]\(x^2 - 9 = 0\)[/tex] has 2 real solutions.
The equation [tex]\(x^2 - 9 = 0\)[/tex] has [tex]\(\boxed{2}\)[/tex] real solution(s).
