How many real solutions does the equation 2sinx = x² - xπ + π²+8/4 have?

Question

How many real solutions does the equation \( 2 \sin x=x^{2}-x \pi+\frac{\pi^{2}+8}{4} \) have?

Answer 1

2sinx=x²-xπ+(π²+8)/4

Sinx=x²/2-xπ/2+(π²+8)/8

Min value of quadratic is = -D/4a

D= b²-4ac

-D/4a = -[(π²/4 - 4(1/2(π²+8)/8)]÷4(1/2)

=1 

As we know that sinx lies between [-1,1]

So there is exactly one solution which is 1