Quirky equation III

$x + \sqrt{x + \dfrac{1}{2} + \sqrt{x+\dfrac{1}{4} } } =1024$

$$

$x + \sqrt{ x + \dfrac{1}{4} + \dfrac{1}{4} + 2\times \dfrac{1}{2} \times \sqrt{x + \dfrac{1}{4} } } = 2^{10}$

$x + \sqrt{ \left ( \sqrt{x + \dfrac{1}{4} } + \dfrac{1}{2} \right)^2 } = 2^{10}$

$x + \dfrac{1}{2} + \sqrt{ x + \dfrac{1}{4} } = 2^{10}$

$$ Splitting the terms the same way again, $\left (\sqrt{ x + \dfrac{1}{4} } + \dfrac{1}{2} \right)^{2} = 2^{10}$

$\sqrt{ x + \dfrac{1}{4}} = 32 - \dfrac{1}{2}$

$$ Squaring,

$$ $x + \dfrac{1}{4} = 1024 -2 \times \dfrac{1}{2} \times 32 + \dfrac{1}{4}$

$$ $x = 1024 - 32 = 992$

$x+ \sqrt{x+ \dfrac{1}{2}+ \sqrt{x+\dfrac{1}{4}}}=1024$

Making a variable change we get the following :

$\sqrt{x+\dfrac{1}{4}}=y$

Squaring both sides and adding $\dfrac{1}{4}$ we get:

$x+ \dfrac{1}{2}=y^2+\dfrac{1}{4}$

From this we get:

$x=y^2- \dfrac{1}{4}$

Substituting this three equations into our original equation we get:

$y^2- \dfrac{1}{4}+ \sqrt{y^2+ \dfrac{1}{4}+ y}=1024$

We factorize inside the root, simplify and finally solve the equation:

$y^2- \dfrac{1}{4}+ \sqrt{(y+ \dfrac{1}{2})^2}=1024$

$y^2- \dfrac{1}{4}+ y+ \dfrac{1}{2}=1024$

$y^2+ y+ \dfrac{1}{4}=1024$

$(y+ \dfrac{1}{2})^2=1024$

$y+ \dfrac{1}{2}=32$

$y=31.5$

Now from the variable change:

$x=y^2- \dfrac{1}{4}$

$x=(31.5)^2- \dfrac{1}{4}$

$x=992$

$x+\sqrt{x+1/2+\sqrt{x+1/4}} = 2^{10}$

Let $x+1/4=t^{2}$

$x+1/2=t^{2}+1/4$

$t^{2}-1/4+\sqrt{(t+1/2)^2}=32^{2}$

$(t+1/2)^{2}=32^2$

$t=63/2$

$x+1/4=3969/4$

$\boxed{x=992}$