Question

Find the following for x^2 - 6x = 27:

Discriminant:

Number of solutions:

Types of solutions:

Solutions: x = { }

a) D > 0, Two Real Solutions, x = { 9, -3 }
b) D = 0, One Real Solution, x = { 3 }
c) D < 0, No Real Solutions, x = { }
d) D = 0, Two Real Solutions, x = { 9, -3 }

Answer 1

The discriminant for the quadratic equation x^2 - 6x = 27 is D = 144, which is positive. Therefore, there are two real solutions: x = 9 and x = -3.

Step-by-step explanation:

To find the discriminant of the quadratic equation x^2 - 6x = 27, we can use the formula D = b^2 - 4ac. Here, a = 1, b = -6, and c = -27. Substituting these values, we have D = (-6)^2 - 4(1)(-27) = 36 + 108 = 144. Since the discriminant is positive, D > 0.

The number of solutions for a quadratic equation depends on the value of the discriminant. If D > 0, there are two real solutions. If D = 0, there is one real solution. If D < 0, there are no real solutions.

In our case, since D = 144 > 0, there are two real solutions. To find the solutions, we can use the quadratic formula: x = (-b ± √D) / 2a. Substituting the values, we have x = (6 ± √144) / 2 = (6 ± 12) / 2. Therefore, the solutions are x = 9 and x = -3.