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=0b. 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.

Answer:See below.Step-by-step explanation:(a) Because the solution led to a square root of a negative number:x^2 -10x+40=0x^2 - 10x = -40   Completing the square:(x - 5)^2 - 25 = -40(x - 5)^2 = -15x =  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.