Quadratic equation 11

αandβare the roots of the quadratic equation x^2-5x-3=0
α+β= 5
αβ= -3
(a) The roots of the required equation:
3α+ 3β= 3(α+β)
= 3(5)
=15
(3α)(3β) = 9αβ
= 9(-3)
= -27
The required equation is
x^2 - 15x - 27 = 0
(b)The roots of the required equation is
[-α/3 + (-β/3)] = -α/3 -β/3
= - (α+β)/3
= - 5/3
(-α/3)(-β/3) = αβ/ 9
= -3/9
= -1/3
m and n are the roots of the quadratic equation 2x^2-7x+4=0
m + n = 7/2
mn= 4/2 = 2
(a) The roots of the required equation
1/m + 1/n = (m+n) / mn
= (7/2) /2
= 7/4
(1/m)(1/n) = 1/mn
= 1/2
The required equation is
x^2 - 7/4 + 1/2 = 0
4x^2 - 7 + 2 = 0
(b) 1/m-3 + 1/n-3
= (n-3)+(m-3) / (m-3)(n-3)
= (m+n-6) / (mn-3n-3m+9)
= (7/2 - 6) / [mn - 3(m+n) +9]
= 19/2 / [1/2 - 3(7/2) +9]
= -19/2
(1/m-3)(1/n-3)
=1 / (m-3)(n-3)
=1 / [mn - 3(m+n) +9]
= 1/ [1/2 - 3(7/2) +9]
= -1
The required equation is
x^2 - (-19/2)x + (-1) = 0
x^2 + 19/2 x - 1 = 0
2x^2 + 19x - 2 = 0