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.