Prove that 2x^2+x+8> 0 for all real values of x.

 is a polynomial of even degree, with a positive  coefficient, meaning that
- it will have exactly one turning point
- that turning point will be a minimum

So, if the y-coordinate of the turning point is positive, then this polynomial will be positive for all real values of x.

At a turning point, the gradient of y will be equal to 0. The gradient of y is given by 
.

To find the turning point, set this equal to 0 and solve for x:
.

Substituting this value into the equation gives
.

Since the minimum point of the equation is greater than 0, the equation will be greater than 0 for all real values of x.