急 ! 急! F4 Quadratic Equations.

If you have not studied linear transformation, then I may use sum and product of root to answer you...
18.)
m+n = 2/5
mn = -1/5
New sum of roots = k(m+n) = 2k/5
New product of roots = k^2(mn) = - k^2 /5
Required equation: 5x^2 - 2kx - k^2 = 0

19.)
p+q = 3/2
pq = -1/2
New sum of roots = k/p + k/q = k(p+q)/pq = (3k/2)/(-1/2) = -3k
New product of roots = (k/p)(k/q) = k^2 /pq = k^2 /(-1/2) = -2k^2
Required equation: x^2 + 3kx - 2k^2 = 0

20a.)
1/α+1/β = (α+β)/αβ = -9/2
1/αβ = 4/2 = 2
(α+β)/αβ = -9/2 => (α+β)(2) = -9/2 => α+β = -9/4
1/αβ = 2 => αβ = 1/2 = 2/4
b.)
The required equation: 4x^2 + 9x + 2 = 0

21a.)
m+n = -b/a
mn = c/a
b.)
New sum of roots
= m^2 - 1 + n^2 - 1
= (m^2 + 2mn + n^2) - 2mn - 2
= (m+n)^2 - 2(c/a) - 2
= b^2 / a^2 - 2c/a - 2
= (b^2 - 2ac - 2a^2)/a^2
New product of roots
= (m^2 - 1)(n^2 - 1)
= (mn)^2 - m^2 - n^2 + 1
= (c/a)^2 - (m^2 + 2mn + n^2) + 2mn + 1
= c^2 / a^2 - b^2 / a^2 + 2c/a + 1
= (c^2 - b^2 + 2ac + a^2)/a^2
Equation required:
a^2 x^2 + (2a^2 + 2ac - b^2)x + (c^2 - b^2 + 2ac + a^2) = 0

22a.)
x^2 - ax + b = x^2 - bx + a
(b-a)x = a-b
x = -1
γ = -1
b.)
Put x = -1,
(-1)^2 - a(-1) + b = 0
1 + a + b = 0
a+b = -1
c.)
α+γ = a
αγ = b
β+γ = b
βγ = a
New sum of roots
= α+β
= (α+γ) + (β+γ) - 2γ
= a + b - 2(-1)
= a+b+2
New product of roots
= αβ
= (αγ)(βγ)/γ^2
= ba/(-1)^2
= ba
Required equation: x^2 - (a+b+2)x + ab = 0