1. ## Simplifying surds

I have placed this equation into the quadratic formula:

$\displaystyle 2x^2 + x(-\sqrt{50} - 2\sqrt{7/2}) + \sqrt{175} = 0$

To get:

$\displaystyle x= \dfrac{-(-\sqrt{50} - 2\sqrt{7/2})\pm\sqrt{(-\sqrt{50} - 2\sqrt{7/2})^2 - 4 \times 2 \times \sqrt{175}}}{2 \times 2}$

Which simplifies to:

$\displaystyle x= \dfrac{\sqrt{50} + \sqrt{14}+\sqrt{\sqrt{2800} - \sqrt{11200}}}{4}$

and

$\displaystyle x= \dfrac{\sqrt{50} + \sqrt{14}-\sqrt{\sqrt{2800} - \sqrt{11200}}}{4}$

Now how do I simplify them to:

$\displaystyle \dfrac{\sqrt{50}}{2}$ and $\displaystyle \sqrt{7/2}$

Can someone show me?

2. I think there is something wrong with your work so far. You're going to get complex numbers, which will never simplify down to the desired results. Double-check your simplification of the discriminant inside the square root, there.

[EDIT]: I also don't think your results at the end are the roots of the polynomial. Try plugging them in and see if they satisfy the original equation.

3. The results of $\displaystyle \dfrac{\sqrt{50}}{2}$ and $\displaystyle \sqrt{7/2}$ are the roots of the polynomial. I have checked.

Maybe I did the expansion of $\displaystyle (-\sqrt{50} - 2\sqrt{7/2})^2$ wrong somewhere and that has led me to complex numbers...

4. Yes, the expansion is wrong.

$\displaystyle (-\sqrt{50} - 2\sqrt{7/2})^2 = 64 + \sqrt{2800}$

5. Then I suggest you check again.

$\displaystyle \frac{\sqrt{50}}{2}= \frac{5}{2}\sqrt{2}$ is approximately 3.53533, $\displaystyle \sqrt{7/2}$ is approximately 1.87083, and $\displaystyle \sqrt{175}= 5\sqrt{3}$ is approximately 13.22876. $\displaystyle 2x^2- (\frac{\sqrt{50}}{2}+ \sqrt{7/2})x+ \sqrt{175}= 0$ is $\displaystyle 2x^2- 16.76409x+ 13.22876= 0$.

Putting x= 3.53533 into that gives -33.54313 while putting x= 1.87083 into it gives 24.74118. Neither satisifies the equation.

If a and b are roots of a polynomial of the form (x- a)(x- b)= 0 then we have $\displaystyle x^2- (a+ b)x+ ab= 0$
Now, with $\displaystyle a= \frac{\sqrt{50}}{2}$ and $\displaystyle b= \sqrt{7/2}$, that would be $\displaystyle x^2- (\frac{\sqrt{50}}+ \sqrt{7/2})x+ \sqrt{175}= 0$. But that is NOT what you have- you have an extra "2" multiplying $\displaystyle x^2$.

6. Originally Posted by HallsofIvy
Then I suggest you check again.

$\displaystyle \frac{\sqrt{50}}{2}= \frac{5}{2}\sqrt{2}$ is approximately 3.53533, $\displaystyle \sqrt{7/2}$ is approximately 1.87083, and $\displaystyle \sqrt{175}= 5\sqrt{3}$ is approximately 13.22876. $\displaystyle 2x^2- (\frac{\sqrt{50}}{2}+ \sqrt{7/2})x+ \sqrt{175}= 0$ is $\displaystyle 2x^2- 16.76409x+ 13.22876= 0$.

Putting x= 3.53533 into that gives -33.54313 while putting x= 1.87083 into it gives 24.74118. Neither satisifies the equation.

If a and b are roots of a polynomial of the form (x- a)(x- b)= 0 then we have $\displaystyle x^2- (a+ b)x+ ab= 0$
Now, with $\displaystyle a= \frac{\sqrt{50}}{2}$ and $\displaystyle b= \sqrt{7/2}$, that would be $\displaystyle x^2- (\frac{\sqrt{50}}+ \sqrt{7/2})x+ \sqrt{175}= 0$. But that is NOT what you have- you have an extra "2" multiplying $\displaystyle x^2$.
Actually this becomes:

$\displaystyle x^2 - (\frac{\sqrt{50}}{2} + \sqrt{7/2})x + \frac{\sqrt{175}}{2} = 0$

Multiplying throughout by 2 then gives:

$\displaystyle 2x^2 - (\sqrt{50}} + 2\sqrt{7/2})x + \sqrt{175} = 0$

7. Here:

Quadratic formula solve equation - Wolfram|Alpha

I have checked 4 times now and the roots are correct.

I just want to know how to simplify it now.

8. $\displaystyle x= \dfrac{\sqrt{50} + \sqrt{14}\pm\sqrt{64+\sqrt{2800} - \sqrt{11200}}}{4}$

$\displaystyle x= \dfrac{5\sqrt{2} + \sqrt{14}\pm\sqrt{64+20\sqrt{7} - 40\sqrt{7}}}{4}$

$\displaystyle x= \dfrac{5\sqrt{2} + \sqrt{14}\pm\sqrt{64-20\sqrt{7}}}{4}$

$\displaystyle x= \dfrac{5\sqrt{2} + \sqrt{14}}{4} \pm \dfrac{\sqrt{64-20\sqrt{7}}}{4}$

$\displaystyle x= \dfrac{5\sqrt{2} + \sqrt{14}}{4} \pm \sqrt{\dfrac{64-20\sqrt{7}}{16}}$

$\displaystyle x= \dfrac{5\sqrt{2} + \sqrt{14}}{4} \pm \sqrt{4-\dfrac{5}{4}\sqrt{7}}$

This is What I got so far