Question

If f(x) = [x]2 + 2[x + 1] – 10, then complete solution of f(x) = 0, where [·] represents the greatest integer function, is

Answer 1

Given: The equation f(x) = [x]^2 + 2[x + 1] – 10

To find: Complete solution of f(x) = 0.

Solution:

• Now we have given the equation as f(x) = [x]^2 + 2[x + 1] – 10

• When f(x) = 0, then [x]^2 + 2[x + 1] – 10 = 0

• Now we know the property of greatest integer function that:

• [X+I]=[X]+I, if I is an integer then we can I separately in the Greatest Integer Function.

• So applying this property, we get:

                 [x]^2 + 2[x] + 2[1] – 10 = 0

• Now greatest integer of [1] is 1.

                 [x]^2 + 2[x] + 2 – 10 = 0

                 [x]^2 + 2[x] - 8 = 0

• Now finding the roots, we get:

                 -2 ± √(2²-4(-8) / 2

                 -2 ± √4+32 / 2

                 -2 ± √36 / 2

                 -2 ± 6 / 2

                 -2-6/2, -2+6/2

                 -8/2, 4/2

                 -4, 2

                 [x] = -4

                 x ∈ [-4,-3)

                 [x] = 2

                 x ∈ [2,3)

Answer:

              So the value of x is  [-4,-3) or [2,3).