Argand Plane

Let
Check for
Then,
So, is a real root of the equation
Let be the other root such that or
consider the interval or
Clearly
By Rolle's theorem has a root in or in
But for all . Thus, ,
for any or , which is a contradiction.
Hence, has no real root other than .

Step 1: Represent points: 1+i → (1,1), -2+3i → (-2,3), (5)/(3)i → (0,(5)/(3)).
Step 2: Slope between (1,1) and (-2,3): m=(3-1)/(-2-1)=-(2)/(3)
Step 3: Slope between (1,1) and (0,(5)/(3)): m=((5)/(3)-1)/(0-1)=-(2)/(3)
Step 4: Since slopes are equal, points are collinear.

Step 1: {Find slopes between points}
Using slope formula: Step 2: {Calculate slopes}
Step 3: {Conclusion}
Since all slopes are the same, the points are collinear.