If one zero of the polynomial 3x^2 + 12x - k is reciprocal of the other , then find the value of k.
+2667 Note that if the roots are reciprocals of each other, they will multiply to 1, meaning \(c \over a \) will equal 1, according to Vieta's.
This means \({-c \over 3} = 1\), meaning \(\color{brown}\boxed{k = -3}\)
+128631 3x^2 + 12x - k = 0
Sum of roots = -12/ 3 = -4
So
x + 1/x= -4
x^2 + 4x + 1 = 0
x^2 + 4x = -1 complete the square on x
x^2 + 4x + 4 = -1 + 4
(x + 2)^2 = 3 take both roots
x + 2 = sqrt (3) x + 2 = -sqrt (3)
x = sqrt (3) - 2 x = -sqrt (3) - 2
Product of the roots = -k /3
So
(sqrt (3) - 2) (-sqrt (3) -2) = -k/3
- (sqrt (3) - 2) (sqrt (3) + 2) = -k/3
(sqrt (3) -2) ( sqrt (3) + 2) = k /3
3 -4 = k/3
-1 = k/3
k = -3
