Question

The solution to the equation x^2 - x - cos y + 1.25 = 0 is

Answer 1

Answer with explanation:

→-1 ≤ cos y≤1

→$[\frac{-\pi}{2}≤y\frac{\pi}{2}]$

The given equation is,

$x^2 - x - cos y + 1.25 = 0\\\\x²-x+1.25=cos y$

→-1≤x²-x+1.25≤1

The two quadratic inequality will be

→ x²-x+1.25+1≥0 ∧ →x²-x+1.25-1≥0

→x²-x+2.25≥0  ∧→ x²-x+0.25≥0

To solve the inequality , we can use ,the method of square or determinant method,

→(x- 0.50)²-0.25+2.25≥0→(x-0.50)²≥-2→As, square of any number is positive, so (x-0.50)², will always be greater than , -2.it means, x∈R.

→(x-0.50)²-0.25+0.25≥0→(x-0.50)²≥0→x-0.50≥0

Square of any number will be greater than or equal to zero.

→x≥0.50 ∨ x≤-0.50

→x∈R

The Solution set is, x∈R, for ,y∈$[\frac{-\pi}{2}, \frac{\pi}{2}]$