**Question 11:**The quadratic equation ${x}^{2}-4x-1=2p(x-5)$ , where

*p*is a constant, has two equal roots. Calculate the possible values of

*p*.

*Solution*:

**Question 12:**Find the range of values of

*k*for which the equation ${x}^{2}-2kx+{k}^{2}+5k-6=0$ has no real roots.

*Solution*:

**Question 13:**Find the range of values of

*p*for which the equation $5{x}^{2}+7x-3p=6$ has no real roots.

*Solution*:

**Question 14:**$\text{Showthat}6x-6-2k{x}^{2}={x}^{2}\text{hasnorealrootsif}k\frac{1}{4}.$

*Solution*:

**Question 15:**The quadratic equation ${x}^{2}+px+q=0$ has roots –2 and 6. Find

(a) the value of

*p*and of

*q*,

(b) the range of values of

*r*for which the equation ${x}^{2}+px+q=r$ has no real roots.

*Solution*: