How do you solve 9 cos^3 x − 63 cos^2 x + cos x − 7 = 0 ?

1 Answer
Oct 25, 2015

No solution.

Explanation:

[1]" "9cos^3x-63cos^2x+cosx-7=0

Pair 9cos^3x and cosx. Pair -63cos^2x and -7.

[2]" "(9cos^3x+cosx)+(-63cos^2x-7)=0

Factor out each group.

[3]" "(cosx)(9cos^2x+1)+(-7)(9cos^2x+1)=0

Factor out 9cos^2x+1 from the entire expression.

[4]" "(9cos^2x+1)(cosx-7)=0

For this to be true, (9cos^2x+1) or (cosx-7) must be equal to 0. However, (cosx-7) will never be 0 because the range of the cosine function is only [-1,1]. Therefore, we can't use that to find x.

[5]" "9cos^2x+1=0

[6]" "9cos^2x=-1

[7]" "cos^2x=-1/9

However, this doesn't make any sense because any real number that is squared must be positive. cos^2x can not be equal to -1/9.

Therefore, we can conclude that the equation has no solution.