How do you factorize −9z2+24z−32 over C?

1 Answer
Andy Y.
May 22, 2017

−9z2+24z−32=(x−43+√153i)(x−43−√153i)

Explanation:

If by "factorize...over C", you mean the set of complex numbers, then here are the following steps:

Step 1. Set the equation equal to zero
−9z2+24z−32=0

Step 2. Plug the coefficients into the quadratic equation
z=−b±√b2−4ac2a where a=−9, b=24, and c=−32

This gives

z=−24±√242−4(−9)(−31)2(−9)

Step 3. Simplify
z=−24±√576−1116−18
z=−24±√−540−18
z=−24±√−36⋅15−18
z=−24±6√−15−18
z=−24±6√15i−18
z=−24−18±6√15i−18
z=43±−√153i

Step 3. Break into two parts

z=43−√153i and z=43+√153i

z−43+√153i=0 and z−43−√153i=0

Step 4. Multiply these terms together for the final answer
−9z2+24z−32=(z−43+√153i)(z−43−√153i)

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