Solution assignment 10 Quadratic equations (abc-formula)

For which value(s) of $p$ does the equation::

$px^2-2x+p+1=0$

have no solutions $x$ .

When we want to know how many solutions a quadratic equation has, we look at the discriminant:

$D=4-4p(p+1)=4-4p^2-4p$

The equation may not have solutions, thus the discriminant has to be negative:

$4-4p^2-4p<0$

or:

$p^2+p-1>0$

We know that the graph of the function:

$f(p)=p^2+p-1$

is an 'opens up' parabola which intersects the $P$ -axis in the points:

$p_{1,2}=\displaystyle\frac{-1\pm\sqrt{1+4}}{2}=\displaystyle\frac{-1\pm\sqrt{5}}{2}$

This parabola is positive if:

$p>\displaystyle\frac{-1+\sqrt{5}}{2}$ or $p<\displaystyle\frac{-1-\sqrt{5}}{2}$