Question 4:
It is given α and β are the roots of the quadratic equation x (x – 3) = 2k – 4, where k is a constant.
(a) Find the range of values of k if α≠β.(b) Given α2 and β2 are the roots of another quadratic equation 2x2+tx−4=0, where t is a constant, find the value of t and of k.
Solution:
(a)x(x−3)=2k−4x2−3x+4−2k=0a=1, b=−3, c=4−2k b2−4ac>0(−3)2−4(1)(4−2k)>0 9−16+8k>0 8k>7 k>78
(b)From the equation x2−3x+4−2k=0,α+β=−ba =−−31 =3.............(1)αβ=ca =4−2k1 =4−2k.............(2)From the equation 2x2+tx−4=0,α2+β2=−t2α+β=−t.............(3)α2×β2=−42αβ=−8.............(4)Substitute (1)=(3),3=−tt=−3Substitute (2)=(4),4−2k=−84+8=2kk=6
It is given α and β are the roots of the quadratic equation x (x – 3) = 2k – 4, where k is a constant.
(a) Find the range of values of k if α≠β.(b) Given α2 and β2 are the roots of another quadratic equation 2x2+tx−4=0, where t is a constant, find the value of t and of k.
Solution:
(a)x(x−3)=2k−4x2−3x+4−2k=0a=1, b=−3, c=4−2k b2−4ac>0(−3)2−4(1)(4−2k)>0 9−16+8k>0 8k>7 k>78
(b)From the equation x2−3x+4−2k=0,α+β=−ba =−−31 =3.............(1)αβ=ca =4−2k1 =4−2k.............(2)From the equation 2x2+tx−4=0,α2+β2=−t2α+β=−t.............(3)α2×β2=−42αβ=−8.............(4)Substitute (1)=(3),3=−tt=−3Substitute (2)=(4),4−2k=−84+8=2kk=6