Question

Consider the quadratic equation x^2-10x=-29. A: is x=5+2i a solution to the equation? how can you be sure without solving?

B: without solving, predict another solution to the equation. verify your prediction by checking it.
C: where does the parabola y=x^2-10x+29 intersect the x-axis? Explain.​

Answer 1

A.

by simply putting the suggested solution into the equation and see if it stays true. if yes, it is a solution.

(5 + 2i)² - 10(5 + 2i) = -29

25 + 20i - 4 - 50 - 20i = -29

21 - 50 = -29

-29 = -29 true

yes, it is a solution.

B. for a parabola the 2 solutions are usually symmetrical around the center line.

so, I suspect 5 - 2i to be a solution too.

(5 - 2i)² - 10(5 - 2i) = -29

25 - 20i - 4 - 50 + 20i = -29

21 - 50 = -29

-29 = -29 true

yes, it is a solution too.

C.

nowhere.

with 2 complex solutions there are no real number solutions left. and that means there is no intersection with the x-axis.

every quadratic equation must have 2 and only 2 solutions. a solution is normally an intersection with the x-axis (the x- value when y = 0).