Need help solving these simultaneous equations, anyone with a pen and paper?

Firepanda
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Solve

-r*(T^2 + T + 6) + 1 = 0

4r*(T^2 + T) - T = 0

For T and r

One solution is T=0 and r=1/6

There are 2 more.


I have T = +- i/(2)0.5 and r = 1/[4(1+T)]

But MATLAB is disagreeing with my solutions, can anyone rectify them for me please?
 
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ah nevermind, foudn my mistake early on, sorry for anyone bothering to attempt it I've done it (after 3 hours :/ )
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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