# No. of roots of $\sin\pi x=x^2-x+{5\over4}$

Find the number of roots of the equation

$\sin\pi x=x^2-x+{5\over4}$

Is there any general formula or rule to find out the number of roots of an equation?

HINT:

$$x^2-x+\frac{5}{4}=(x-\frac{1}{2})^2+1\ge 1$$

So how many intersection can there be, if any, and where?

Plot the 2 functions and you should see that they never intersect. Study the functions and you will find interesting things on the maximas / minimas

Since $\sin\pi x-x^2+x-\dfrac{5}{4}$ is an entire function,

By the principle in properties about number of solutions of transcendental equations, $\sin\pi x=x^2-x+\dfrac{5}{4}$ should have infinitely many solutions (include complex solutions).

Only $$1$$ root.... Because we know $$\sin\text{(something)}$$ has maximum value $$1$$..., so the only root occurs when $$x^2+x-\dfrac{5}{4}=1$$... i.e. $$x = \dfrac{1}{2}$$ which is the one and only root.