I have a number in a + bi form, the number is $25 \pm 48.73397172i$. I desperately need to find a way to get this number into simplest radical form. I am trying to write a program for a TI-83+ calculator that allows me to take a quadratic formula problem (the sample I am using is $5p^2 - 10p + 24 = 0$). I can get it into $a + bi$ form using a simple quadratic formula program I wrote a while back, but I just can't seem to figure out how to turn it into simplest radical form. Any help is appreciated whether it's help with a calculator program or with turning my $a + bi$ solution into a simplest radical form.
P.S. Simplest radical form should be $\dfrac{5 \pm i\sqrt{95}} {5}$
$25 \pm 48.73397172i\approx 5 (5 \pm i\sqrt{95})$, not $\frac{5 \pm i\sqrt{95}}5$.
As DonAntonio did, square $48.73397172$ and you get a value very close to $2375$. (The square root of an integer is either an integer itself or is irrational. Since $\sqrt{2375}$ is obviously not integer, it is irrational. And because it is irrational, it cannot be expressed in any finite number of digits. Thus your $48.73397172$ is only an approximation of the actual square root.)
So your values are actually $25 \pm \sqrt{2375}i$. But the square root can be reduced. Factor $2375 = 5^3\cdot19$, so $\sqrt{2375} = \sqrt{5^2}\sqrt{5\cdot19} = 5\sqrt{95}$.
So your values can be expressed as $25 \pm 5\sqrt{95}i$. But there is a common multiple of 5 between the terms, which can be factored out, leaving them as
$$5 (5 \pm i\sqrt{95})$$