Solutions to a quadratic given 1 solution in form a+bi

Question

I was just really confused as to how I only ended up with 1 of 2 answers for the following question.

Given that $-2+bi$ is a solution of $x^2+ax+(3+a) $ find constants $a$ and $b$ given that they are real.

As soon as I saw that $-2+bi$ was a solution, I immediately jumped to $ -2-bi$ must also be a solution, by the fundamental theorem of algebra. By doing the sum and product of the solution a quadratic could be obtained

Sum $ (-2+bi)+(-2-bi)=-4$

Product $(-2-bi)(-2+bi)=(-2)^2-(-bi)^2=4+b^2$

Thus the quadratic $x^2+4x+(4+b^2)$ is obtained

Equation both sides of the equation

$\\x^2+4x+(4+b^2)=x^2+ax+(3+a)$

$a=4$

Therefore$ b={\sqrt 3}$ or $ b={-\sqrt 3}$

However, the solutions seem to suggest that an extra solution can be $ b={0}, a=7$

Did I eliminate a solution by doing the sum and product of the solutions to find the quadratic? Or does it have to do with my working process

Answer 1

You missed a solution when you assumed that $-2-bi$ was the other root. If $b \neq 0$, then yes, that must be the other root. But if $b = 0$, the other root can be whatever other real number it needs to be. (Also, the fundamental theorem of algebra does not say that non-real roots come in complex conjugate pairs.)

Answer 2

A brute force approach is to substitute $x=-2+bi$ into $x^2+ax+(3+a)$. Then \begin{align} x^2+ax+(3+a) &= (-2+bi)^2+a(-2+bi)+(3+a) \\ &=(7-a-b^2)+b(a-4)i \\ \end{align} where I have organized the last expression into real and imaginary parts (since $a,b$ are assumed real). If this last expression is to vanish as desired, then both real and imaginary parts are zero. From the latter, we deduce that $b=0$ or $a=4$. If $b=0$, then the condition $7-a-b^2=0$ yields $a=7$. If $a=4$, then this same condition implies $b^2=7-a=3$ and therefore $b=\pm \sqrt{3}$. This gives all three solutions.

Answer 3

Let the second root be $c-ib$ (because $2+ib+c-ib=a$ must be real). Then

$$(x+2-ib)(x-c+ib)=x^2+(2-c)x-2c+b^2+i(bc+2b)=x^2+ax+a+3.$$

For the imaginary term to vanish, $$b=0\lor c=-2$$

gives us the three solutions

$$a=7,b=0,c=-5$$ and $$a=4,b=\pm\sqrt3,c=-2.$$