The number of values of the parameter α ∈ 0 , 2 π for which the quadratic function sin ⁡ α x 2 + 2 cos ⁡ α x + 1 2 cos ⁡ α + sin ⁡ α is the square of a linear function is

Step-by-step solution

For the quadratic function to be square of a linear function D = 0 ⇒ 4 cos 2 ⁡ α - 4 · 1 2 sin ⁡ α cos ⁡ α + sin ⁡ α = 0 ⇒ 2 cos ⁡ α + sin ⁡ α cos ⁡ α - sin ⁡ α = 0 ⇒ tan ⁡ α = 1 or - 2 For α ∈ 0 , 2 π there are two solutions for each. ⇒ total four values of α .