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

Question

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?

Answer 1

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

Answer 2

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?

Answer 3

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).

Answer 4

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.