discussion-back-with-mr-W-consider-this-equation-x-2-2x-3-3-x-27-0-does-the-equation-have-two-roots-or-three-roots-

Question Number 79647 by john santu last updated on 27/Jan/20

discussion back with mr W .

consider this equation

(x^{2} - 2 x - 3) (3^{x} - 27) = 0

does the equation have two roots

or three roots ?

Commented by naka3546 last updated on 27/Jan/20

x \in R o r x \in C ?

Commented by john santu last updated on 27/Jan/20

x \in R sir

Answered by MJS last updated on 27/Jan/20

x^{2} - 2 x - 3 = 0 \lor 3^{x} - 27 = 0

(x = - 1 \lor x = 3) \lor (x = 3) \equiv (x = - 1 \lor x = 3)

2 roots

we have different kinds of roots, in this case

1 single root x = - 1

1 double root x = 3

Commented by MJS last updated on 27/Jan/20

f^{'} (x) = 2 (x - 1) (3^{x} - 27) + (x^{2} - 2 x - 3) 3^{x} \ln 3

f^{'} (- 1) = \frac{320}{3} > 0 \Rightarrow {\begin{cases} f (x) < 0; x < - 1 \\ f (x) > 0; x > - 1 \end{cases}

f^{'} (3) = 0 \Rightarrow f (x) > 0; - 1 < x < \infty

at x = 3 {there}^{'} s also a local minimum

counting roots means to count the x - values

at which y becomes 0

after counting we classify these roots

y = {(x - 3)}^{17} has only one root but {it}^{'} s a

seventeen - fold root

Commented by john santu last updated on 27/Jan/20

ok . thank you mister