1. Find a quadratic equation whose roots are 3 and 5.

2. For what value of [tex]$\lambda$[/tex] will the quadratic equation [tex]$x^2 - (\lambda+3)x + 9 = 0$[/tex] have equal roots?

3. If one root of the equation [tex]$x^2 - px + q = 0$[/tex] is twice the other, show that [tex]$2p^2 = 9q$[/tex].

4. Find the value of [tex]$k$[/tex] so that the equation [tex]$2x^2 + (4-k)x - 17 = 0$[/tex] has roots that are equal but opposite in sign.

5. Discuss the nature of the roots of the equations:
a. [tex]$3x^2 + x - 2 = 0$[/tex]
b. [tex]$2x^2 + 2x + 1 = 0$[/tex]
c. [tex]$\dot{x}^2 + 2x + 1 = 0$[/tex]
d. [tex]$4x^2 - 4x + 1 = 0$[/tex]

6. Find the value of [tex]$k$[/tex] so that the equation [tex]$7x^2 - 21x - k = 0$[/tex] has reciprocal roots.

7. If [tex]$\alpha$[/tex] and [tex]$\beta$[/tex] are the roots of [tex]$px^2 + x + q = 0$[/tex], prove that
[tex]$\sqrt{\frac{\alpha}{\beta}} + \sqrt{\frac{\beta}{\alpha}} + \sqrt{\frac{q}{p}} = 0.$[/tex]