Answer:
Number of solutions: 0
Type of solutions: nonreal (aka nondistinct)
Explanation:
Convert x^2 - 6x + 5 = -11 to standard form:
Before we can determine the number and type of solutions, we need the quadratic to be in standard form, whose general equation is given by:
ax^2 + bx + c = 0
We can put the quadratic in standard form by adding 11 to both sides:
(x^2 - 6x + 5 = -11) + 11
x^2 - 6x + 16 = 0
Thus, 1 is our a value, -6 is our b value, and 16 is our c value.
Using the discriminant to determine the type and number of solutions:
- The discriminant (D) comes from the quadratic formula and it is given by the formula D = b^2 - 4ac.
- For any quadratic, there can be either 0, 1, or 2 solutions and a solution can either be real (aka distinct) or nonreal.
There are three things the discriminant can tell us about the number and type of solutions for a quadratic:
- When D < 0, there are 0 real solutions.
- When D = 0, there is 1 real solution.
- When D > 0, there are 2 real solutions.
Since we know that a = 1, b = -6, and c = 16, we can use the discriminant to determine the type and number of solutions:
D = (-6)^2 - 4(1)(16)
D = 36 - 64
D = -28
Since -28 is less than 0, there are 0 real solutions.