Question

Solve the following equation by using formula :

a (x² + 1) = (a²+ 1) x , a ≠ 0

Answer 1

a (x² + 1) = (a²+ 1) x 
ax²– (a² + 1)x + a = 0
Here a = a, b = -(a² + 1), c = a
D = b² - 4ac
= [-(a² + 1)]² - 4 x a x a
= a⁴ + 2a² + 1 - 4a²
= a⁴ - 2a + 1
= (a² - 1)²

∵ x = `(-b ± sqrt("D"))/(2a)`

= `((a^2 + 1) ± sqrt((a^2 - 1)^2))/(2a)`

= `((a^2 + 1) + (a^2 - 1))/(2a)`

∴ x₁ = `(a^2 + 1 + a^2 - 1)/(2a)`

= `(2a^2)/(2a)`
= a

x₂ = `(a^2 + 1 - a^2 + 1)/(2a)`

= `(2)/(2a)`

= `(1)/a`

Hence x = `a, (1)/a`.