Answer: To find the number and type of solutions for the given equation, we can use the discriminant, which is the part of the quadratic formula that determines the number and type of solutions. The discriminant for a quadratic equation in the form ax^2 + bx + c = 0 is b^2 - 4ac.
In the case of the given equation x^2 - 9x + 8 = 0, we can substitute the values of a, b, and c to obtain the discriminant:
discriminant = b^2 - 4ac = (-9)^2 - 4 * 1 * 8 = 81 - 32 = 49
The value of the discriminant tells us the number and type of solutions for the given equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one double real solution. And if the discriminant is negative, the equation has no real solutions.
In the case of the given equation, the discriminant is 49, which is positive. This means that the equation has two distinct real solutions. To find these solutions, we can use the quadratic formula:
x = (-b ± √discriminant) / (2a)
Substituting the values for a, b, c, and the discriminant, we obtain:
x = (-9 ± √49) / (2 * 1) = (-9 ± 7) / 2
This gives us two solutions: x = (-9 + 7) / 2 = -1 and x = (-9 - 7) / 2 = -8. Thus, the given equation has two distinct real solutions: x = -1 and x = -8.