1. Find the set of values of 𝑥 or find the range of values of 𝑥 which satisfy the inequalities:
a) 𝑥(𝑥 − 2) < 3
e) (𝑥 − 1)(5𝑥 + 4) ≤ 2(𝑥 − 1)
b) 𝑥 2 + 5𝑥 − 6 < 0
c) 𝑥 2 > 4𝑥 + 12
d) 4𝑥(𝑥 + 1) ≤ 3
g) (𝑥 + 2)2 < 𝑥(4 − 𝑥) + 40
2. The roots of the equation (𝑝 + 2)𝑥 2 − 2𝑝𝑥 = 5 − 𝑝 are not real; find the range of values of 𝑝.
3. Find the range of values of 𝑘 for which the equation 2𝑥 2 + 5𝑥 + 3 − 𝑘 = 0 has two real distinct roots.
4. Find the set of values of the constant k for which the equation 2𝑥 2 − 𝑘𝑥 + 2𝑘 = 6 has real roots.
5. Given that the equation 𝑥 2 − 12𝑥 − 6 = 0 has roots 𝛼 and 𝛽.
Find
a) 𝛼 + 𝛽 and 𝛼𝛽
b) (𝛼 + 𝛽)2
c) (𝛼 − 𝛽)2
6. Given that the roots of the equation 3𝑥 2 + 𝑝𝑥 + 4 = 0 are 𝛼 and 3𝛼. Find the two possible values of 𝑝.
7. Given that the equation 𝑥 2 − 2𝑥 − 3 = 𝑝, where 𝑝 is a constant, has repeated roots. Find the value of 𝑝.
8. Given that the equation 4𝑥 2 + 𝑝𝑥 + 2𝑝 = 15, where 𝑝 is a positive constant. Find the values of 𝑝 for which
this equation has equal roots.
9. Complete the square of the following expressions:
a) 𝑥 2 − 9𝑥 − 72
b) 2𝑥 2 − 10𝑥 − 19