Find the value of k for which the roots of the quadratic equation kx(x - 2) + 6 = 0 are equal. *If –4 is a root of the equation x² + px - 4 = 0 and the equation x² + px + q = 0 has equal roots, find the values of p and q. *
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Answer:
x
2
+px+1=0
a+b=−p
ab=1
x
2
+qx+1=0
c+d=−q
cd=1
(a−c)(b−c)=ab−(a+b)c+c
2
(1+pc+c
2
)
(a+d)(b+d)=ab+(a+b)d+d
2
=(1−pd+d
2
)
∴(1+pc+c
2
)(1−pd+d
2
)
=1−pd+d
2
+pc−p
2
cd+c
2
−pc
2
d+c
2
d
2
=1+c
2
+d
2
+pc−pc
2
d+pcd
2
−pd+p
2
cd+c
2
d
2
=1+c
2
+d
2
+pc−pc+pd−pd+p
2
+1
=2+p
2
+(c+d)
2
−2d
=2+p
2
+q
2
−2
=p
2
+q
2
∴
q
2
−p
2
(a−c)(b−c)(a+d)(b+d)
=
q
2
−p
2
q
2
+p
2
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