Find in its simplest form the quadratic equation, whose roots are negative four plus five 𝑖 and negative four minus five 𝑖.
If these are our roots, we can set them equal to 𝑥 and then we can move them over to the left with 𝑥 so it’s equal to zero. So when working backwards, we have these factors and we brought them over to the left to be with 𝑥. Now we can take these two and multiply them together because that’s what factors do; they multiply together to be our equation.
And now we FOIL. First, we distribute 𝑥, then we distribute four, and then finally we distribute negative five 𝑖. Now we combine like terms. The five 𝑖𝑥s cancel; the 20𝑖s cancel. So we have 𝑥 squared plus eight 𝑥 plus 16 minus 25𝑖 squared.
Now 𝑖 squared is equal to negative one. So it’s really negative 25 times negative one. So it is actually 25. So we can combine like terms: the 16 and the 25. So our quadratic equation would be 𝑥 squared plus eight 𝑥 plus 41.