Question

If alpha and beta are the zeroes of the polynomial f(x)=Kx²+4x+4 such that alpha square + beta square is 24.Find the value of K.

Answer 1

Given alpha and beta are the zeroes of the polynomial kx^2+4x+4 I 'll denote alpha by x and beta by y for the sake of better to type and simplicity f(x)=ax²+bx+c (general equation) equation f(x)=Kx²+4x+4 sum of roots, x + y = -b/a (here b = -4 and a = 1) x+y=-4/k.......eq 1 product of roots , xy= c/a ( c = 4 and a = 1) xy= 4/k........eq 2 Squaring both sides of eq 1 (x+y)2= (-4/k)² x²+y²+2xy=16/k² 24+2xy=16/k² [from x²+y² = 24] Substituting the value of xy from eq2, 24+2*4/k=16/k² 24+8/k=16/k² 24k+8/k=16/k² 24k+8=(16/k²)*(k) 24k+8=16/k 24k²+8k=16 or 24k²+8k -16=0 3k²+k-2=0 3k²+3k-2k-2=0 3k(k+1)-2(k+1)=0 (3k-2)(k+1)=0 Therefore,k is either -1 or 2/3
Zeroes is otherwise known as roots of the equation.