Solve the equation $\sqrt{x^2+x+1}+\sqrt{x^2+\frac{3x}{4}}=\sqrt{4x^2+3x}$

Question

Solve the equation $$\sqrt{x^2+x+1}+\sqrt{x^2+\dfrac{3x}{4}}=\sqrt{4x^2+3x}$$
The domain is $$x^2+\dfrac{3x}{4}\ge0,4x^2+3x\ge0$$ as $x^2+x+1>0$ for every $x$. Let's raise both sides to the power of 2: $$x^2+x+1+x^2+\dfrac{3x}{4}+2\sqrt{(x^2+x+1)\left(x^2+\dfrac{3x}{4}\right)}=4x^2+3x\\2\sqrt{(x^2+x+1)\left(x^2+\dfrac{3x}{4}\right)}=2x^2+\dfrac{5x}{4}$$ Let's raise both sides to the power of 2 again but this time the roots should also satisfy $A:2x^2+\dfrac54x\ge0$:$$4(x^2+x+1)\left(x^2+\dfrac{3x}{4}\right)=(2x^2+\dfrac54x)^2$$ I came at $$x(2x^2+\dfrac{87}{16}x+3)=0$$ I obviously made a mistake as the answer is $x=-4$, but is there an easier approach?

Answer 1

HINT

I would start with multiplying both sides by the number $2$: \begin{align*} \sqrt{x^{2} + x + 1} + \sqrt{x^{2} + \frac{3x}{4}} = \sqrt{4x^{2} + 3x} & \Longleftrightarrow 2\sqrt{x^{2} + x + 1} + \sqrt{4x^{2} + 3x} = 2\sqrt{4x^{2} + 3x}\\\\ & \Longleftrightarrow 2\sqrt{x^{2} + x + 1} = \sqrt{4x^{2} + 3x}\\\\ \end{align*}

Can you take it from here?