Question

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.

Answer 1

See below.

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.