math 急...(10點) 快而準者可即得10點.

1. In each pair of the following roots, form a quadratic equation in x.
a. 5-2√3 , 5+2√3
Sum of roots = 10
Product of roots = 25 - 4(3) = 13
Equation is x^2 - 10x + 13 = 0
b. -0.3 , 0.3
Sum of roots = 0
Product of roots = -0.09
Equation is x^2 - 0.09 = 0
100x^2 - 9 = 0
2. Given that alpha;and beta;are the roots of the equation (x + 4)(2x - 1) = 5. Find the value of each of the following expression.
(x + 4)(2x - 1) = 5
2x^2 + 8x - x - 4 = 5
2x^2 + 7x - 9 = 0
α + β = -7/2
αβ = -9/2
a. (α + 1/β)(β + 1/α)
= αβ + 1 + 1 + 1/(αβ)
= -9/2 + 2 - 2/9
= (-81 + 36 - 4)/18
= -49/18
b. α^3 + β^3
= (α + β)(α^2 - αβ + β^2)
= (-7/2)(α^2 + 2αβ + β^2 - 3αβ)
= (-7/2)[(α + β)^2 - 3αβ]
= (-7/2)(49/4 + 27/2)
= (-7/2)(49/4 + 54/4)
= -(7/2)(103/4)
= -721/8
3. Given that α and α - 2 are the roots of the equation x^2 + (3k - 2)x + 48 = 0.
a. Find the value of α
Product of roots α(α - 2) = 48
α^2 - 2α - 48 = 0
(α - 8)(α + 6) = 0
α = 8 or α = -6
b. If α＞0, find the value of k
α = 8
Sum of roots = 8 + (8 - 2) = 14 = -(3k - 2)
3k = -14 + 2
k = -4
4. Form a quadratic equation whose roots are 4 + 3√2 and 4 - 3√2
Sum of roots = 8
Product of roots = 16 - (9)(2) = -2
Equation is x^2 - 8x - 2 = 0.
5. Suppose α and β are the roots of the equation x^2 - 7x - 3 = 0 . Form a quadratic equation with roots α/βand β/α .
αβ = -3
α + β = 7
Sum of roots α/β + β/α
= (α^2 + β^2) / αβ
= (α^2 + β^2 + 2αβ - 2αβ) / αβ
= [(α + β)^2 - 2αβ] / αβ
= (49 + 6) / -3
= -55/3
Product of roots = (α/β)(β/α) = 1
Equation is x^2 + 55x/3 + 1 = 0
3x^2 + 55x + 3 = 0