Solve 𝑧 squared plus two minus two 𝑖 𝑧 minus seven plus 26𝑖 equals zero.
Here we have a quadratic equation. This time though, it has nonreal coefficients. This doesn’t really matter though. It means only to be a little bit careful. But we can still apply the quadratic formula.
Remember, for an equation of the form 𝑎𝑧 squared plus 𝑏𝑧 plus 𝑐 equals zero, the solutions are given by 𝑧 equals negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all over two 𝑎. And so for our equation, 𝑎, the coefficient of 𝑧 squared, is equal to one. 𝑏 is the coefficient of 𝑧. So that’s two minus two 𝑖. And 𝑐 is the constant term, so that’s negative seven plus 26𝑖.
Let’s substitute everything we have into our quadratic formula. Negative 𝑏 is negative two minus two 𝑖. We then have plus or minus the square root of 𝑏 squared minus four 𝑎𝑐. Remember, 𝑏 squared minus four 𝑎𝑐 is the discriminant. And here that’s two minus two 𝑖 all squared minus four times one times negative seven plus 26𝑖. And this is all over two 𝑎. So that’s just two times one.
Let’s distribute our parentheses. Negative two minus two 𝑖 is negative two plus two 𝑖. When we square two minus two 𝑖, we get four minus eight 𝑖 plus four 𝑖 squared. And negative four times one times negative seven plus 26𝑖 is just 28 plus 104𝑖. And this is all over two.
Now we can simplify a little bit further. We know that 𝑖 squared is equal to negative one. So we can replace four 𝑖 squared with negative four. And then we see that four minus four is zero. And so our solution simplify to negative two plus two 𝑖 plus or minus the square root of 28 plus 96𝑖 over two. But how do we find the square root of a complex number?
Well, let’s say the solution to this is 𝑥 equals 𝑎 plus 𝑏𝑖. 𝑥 squared is 𝑎 plus 𝑏𝑖 all squared, which when we distribute the parentheses becomes 𝑎 squared minus 𝑏 squared plus two 𝑎𝑏𝑖. We know that 28 plus 96𝑖 must be equal to this value, 𝑎 squared minus 𝑏 squared plus two 𝑎𝑏𝑖. And remember of course 𝑎 and 𝑏 are real constants.
We can now equate coefficients. Let’s compare the coefficients of the real part. So on the left, that’s 28. And on the right, that’s 𝑎 squared minus 𝑏 squared. So 28 equals 𝑎 squared minus 𝑏 squared.
We’ll now compare the imaginary parts. On the left, that’s 96. And on the right, that’s two 𝑎𝑏. And so 96 equals two 𝑎𝑏. Let’s rearrange this second equation to make 𝑏 the subject. We divide through by two 𝑎. And we find that 𝑏 is equal to 48 over 𝑎. We then substitute this into our other equation. So we get 28 equals 𝑎 squared minus 48 over 𝑎 all squared. Next, we square 48 over 𝑎 and then multiply this entire equation by 𝑎 squared.
Gathering all the terms together, and we get 𝑎 to the fourth power minus 28𝑎 squared minus 2304 equals zero. Factoring this expression, we get 𝑎 squared minus 64 times 𝑎 squared plus 36. And that’s equal to zero. Now what this actually means is either 𝑎 squared minus 64 is equal to zero. And thus 𝑎 squared is equal to 64. Or 𝑎 squared plus 36 is equal to zero. And 𝑎 squared is equal to negative 36.
In fact though, we disregard this solution because we said that 𝑎 had to be a real constant. If we find the square root of negative 36, we have a nonreal number. So we’re actually only interested in 𝑎 squared equals 64, which means that 𝑎 is equal to plus or minus eight. Substituting in this back into our expression for 𝑏, we get 48 over plus or minus eight, which is equal to plus or minus six.
Now what we could do is replace the square root of 28 plus 96𝑖 with plus or minus eight plus or minus six 𝑖. But actually, we really don’t need the plus or minuses now. And that’s because we’re going to find plus or minus eight plus six 𝑖. So all options are really included here.
Let’s clear some space and perform the last steps. We’ll separate our two solutions. We have negative two plus two 𝑖 plus eight plus six 𝑖 over two or negative two plus two 𝑖 minus eight plus six 𝑖 over two. On the left, that simplifies to six plus eight 𝑖 over two. And on the right, it becomes negative 10 minus four 𝑖 over two. All that’s left to do is to divide everything through by two. And we obtain our two solutions. They are 𝑧 equals three plus four 𝑖 or negative five minus two 𝑖.