<s>[INST] Solve $\sqrt{x^2+8x+7}+\sqrt{x^2+3x+2}=\sqrt{6x^2+19x+13}$ I tried squaring both sides but doesn't seem like a good idea.
$$x^2+8x+7+\sqrt{(x^2+8x+7)(x^2+3x+2)}+x^2+3x+2=6x^2+19x+13$$
$$\sqrt{(x^2+8x+7)(x^2+3x+2)}=4x^2+8x+4$$
$$\sqrt{(x^2+8x+7)(x^2+3x+2)}=4(x+1)^2$$
Is there a better way for solving this equation?
 [/INST] I would factor both sides first so that:
$$\sqrt{(x+7)(x+1)}+\sqrt{(x+2)(x+1)}=\sqrt{(6x+13)(x+1)}$$
$$\sqrt{x+1}(\sqrt{x+7}+\sqrt{x+2})=\sqrt{x+1}\sqrt{6x+13}$$
$$\sqrt{x+7}+\sqrt{x+2}=\sqrt{6x+13}$$
$$2x+9+2\sqrt{(x+2)(x+7)}=6x+13$$
$$\sqrt{(x+2)(x+7)}=2x+2$$
$$(x+2)(x+7)=4x^2+8x+4$$
$$3x^2-x-10=0$$
$$(3x+5)(x-2)=0$$
So $x=-\frac{5}{3}$ or $x=2$. However, you find that only $x=2$ works by putting the value into the equation. Do note that while this method may be easier, another solution, $x=-1$ has not been found which would be found by squaring both sides: https://www.symbolab.com/solver/step-by-step/%5Csqrt%7Bx%5E%7B2%7D%2B8x%2B7%7D%2B%5Csqrt%7Bx%5E%7B2%7D%2B3x%2B2%7D%3D%5Csqrt%7B6x%5E%7B2%7D%2B19x%2B13%7D. While this may not be a complete answer, I hope this helped!
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