  ### Home > CCA2 > Chapter 8 > Lesson 8.3.2 > Problem8-139

8-139.

Spud has a problem. He knows that the solutions for a quadratic equation are $x = 3 + 4i$ and $x= 3 − 4i$, but in order to get credit for the problem he was supposed to have written down the original equation. Unfortunately, he lost the paper with the original equation on it. Luckily, his friends are full of advice.

1. Just remember when we made polynomials. If you wanted 7 and 4 to be the answers, you just used $(x - 7)(x - 4)$. So you just do $x$ minus the first one times $x$ minus the other.” Use $(x − (3 + 4i))(x − (3 − 4i))$ to find the quadratic equation.

2. Hugo says, “No, no, no. You can do it that way, but that's too complicated. I think you just start with $x = 3 + 4i$ and work backward. So $x - 3 = 4i$, then, hmmm... Yeah, that'll work.” Try Hugo's method.

3. Whose method do you think Spud should use? Explain your choice.

$y = (x − (3 + 4i))(x − (3 − 4i))$

2 by 2 rectangle labeled as follows: top edge, x, minus open parenthesis, 3, + 4, I, close parenthesis, left edge, x, minus open parenthesis, 3, minus 4, I, close parenthesis.

Labels added to the interior as follows: top left, x squared, top right, negative, x, times, open parenthesis, 3, + 4, I, close parenthesis, bottom left, negative, x, times, open parenthesis, 3, minus 4, I, close parenthesis, bottom right, open parenthesis, 3, minus 4, I, close parenthesis, times, open parenthesis, 3, + 4, I, close parenthesis.

Labels changed to the interior as follows: top right, negative 3, x, minus 4, x, I, bottom left, negative 3, x, + 4, x, I, bottom right, 25

$y = x^2 - 6x + 25$   $(x - 3)^2 = (4i)^2$