Video: Forming a Quadratic Equation in the Simplest Form given Its Roots Involving Complex Numbers

Find in its simplest form, the quadratic equation whose roots are βˆ’4 + 5𝑖 and βˆ’4 βˆ’ 5𝑖.

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Video Transcript

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 π‘₯ and then we can move them over to the left with π‘₯ 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 π‘₯. 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 π‘₯, then we distribute four, and then finally we distribute negative five 𝑖. Now we combine like terms. The five 𝑖π‘₯s cancel; the 20𝑖s cancel. So we have π‘₯ squared plus eight π‘₯ 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 π‘₯ squared plus eight π‘₯ plus 41.

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