Factorize 50. (Complex Algebra)

First, we have to simplify $x = \dfrac{50}{8i+6}$

$x = \dfrac{50}{8i+6}$

$= \dfrac{50}{2(3+4i)}$

$= \dfrac{25}{(3+4i)}$ $~~ ~~ ~ ~ ~~ ~ ~$ $; \left[ \dfrac{50}{2} = 25 \right]$

$= \dfrac{25(3-4i)}{(3+4i)(3-4i)}$ $~~ ~ ~~ ~ ~ ~$ $; \left[\text{Multiply by} \dfrac{3-4i}{3-4i} \right]$

$= \dfrac{25(3-4i)}{25}$

$= 3-4i$

Now $a + bi,$ where $a = 3$ and $b =4$

Hence $a + b = 3+ (-4) = \boxed{-1}$

$\begin{aligned} x & = \frac {50}{8i+6} \\ & = \frac {50(6-8i)}{(6+8i)(6-8i)} \\ & = \frac {50(6-8i)}{36+64} \\ & = 3-4i \end{aligned}$

$\implies a+b = 3-4= \boxed{-1}$

$(8i+6) \times (3-4i) = 50$

$(8i+6) \times (3-4i) = (3 \times 8i) + (-4i \times 6) + (3 \times 6) + (-4i \times 8i) = 24i - 24i + 18 + 32 = 50$

$\frac{(8i+6) \times (3-4i)}{(8i+6)}= 3-4i$

$3-4i \equiv a+bi$

$a = 3$ , $b = -4$

$3 + (-4) = -1$

$\boxed{-1}$