1. for centuries, mathematicians believed that quadratic equations, like the one below, had no solutions and were not solvable.

a. why did they believe this? explain your answer.
x^2 -10x+40=0

b. using the concept of i, complete the problem above and find the two complex solutions.

c. substitute your value into your equation to prove that your solutions found in part b are correct.

See below.

Step-by-step explanation:

(a) Because the solution led to a square root of a negative number:

x^2 -10x+40=0

x^2 - 10x = -40   Completing the square:

(x - 5)^2 - 25 = -40

(x - 5)^2 = -15

x =  5 +/-√(-15)

There is no real square root of -15.

(b) A solution was found by introducing the operator i which stands for the square root of -1.

So the solution is  

= 5 +/- √(15) i.

These are called complex roots.

(c) Substituting in the original equation:

x^2 - 10 + 40:

((5 + √(-15)i)^2 - 10(5 + √(-15)i) + 40

= 25 + 10√(-15)i - 15 - 50 - 10√(-15)i  + 40

=  25 - 15 - 50 + 40

= 0.   So this checks out.

Now substitute 5 - √(-15)i

= 25 - 10√(-15)i - 15 - 50 + 10√(-15)i  + 40

=  25 - 15 - 50 + 40

= 0.  This checks out also.