Question

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. *

Answer 1

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