Express a number in the form $a + b\sqrt{c}$

Question

I have looked into solutions of completing the square however the only example questions I can find are such as $$4x^2 – 2x – 5$$ and so on.

In an exam coming up I will have a question like this :

Given that c = 13, find the values of a and b below :

$$\sqrt{1573}(6-\sqrt{5200}) = a + b\sqrt{c}$$

I know that I have to simplify the numbers into their primes to begin with but how do I do that ?

Answer 1

It has nothing to do (directly) with quadratic equations.

Hint:

$1573=11^2\cdot 13$, $\;5200=20^2\cdot 13$.

Some details: $$\sqrt{1573}(6-\sqrt{5200})=11\sqrt{13}(6-20\sqrt{13})=66\sqrt{13}-220\cdot\sqrt{13}^2=\underbrace{-2860}_{\vphantom{b}a}+\underbrace{66}_b\sqrt{13}. $$

Answer 2

Starting with $\sqrt{1573}(6-\sqrt{5200}) = a + b\sqrt{c} $, first, factor the numbers and take any squares out of the square roots.

Since $1573 =11^2\cdot 13 $ and $5200 =10^22^213 $, this becomes $11\sqrt{13}(6-20\sqrt{13}) = a + b\sqrt{c} $.

By the magic of homework problems, the values inside the square roots are the same.

Multiplying this out, $66\sqrt{13}-220\cdot 13 = a + b\sqrt{c} $, or, rearranging, $-2860+66\sqrt{13} =a+b\sqrt{c} $. And we are done.