"The sum, of the squares of all the roots of the equation $x^2+|2 x-3|-4=0$, is"

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To find the sum of the squares of all the roots of the equation $ x^2 + |2x - 3| - 4 = 0 $:

For $ x \geq \frac{3}{2} $, the expression $ |2x - 3| $ becomes $ 2x - 3 $. Thus, the equation becomes:

$ x^2 + 2x - 3 - 4 = 0 $

Simplifying gives:

$ x^2 + 2x - 7 = 0 $

Solving this quadratic equation, we find:

$ x = 2\sqrt{2} - 1 $

For $ x < \frac{3}{2} $, the expression $ |2x - 3| $ becomes $ -(2x - 3) = -2x + 3 $. The equation therefore becomes:

$ x^2 + 3 - 2x - 4 = 0 $

Simplifying gives:

$ x^2 - 2x - 1 = 0 $

Solving this quadratic equation, we obtain:

$ x = 1 - \sqrt{2} $

The sum of the squares of the roots is:

$ (2\sqrt{2} - 1)^2 + (1 - \sqrt{2})^2 $

Calculating each term:

$(2\sqrt{2} - 1)^2 = (2\sqrt{2})^2 - 2 \cdot 2\sqrt{2} \cdot 1 + 1^2 = 8 - 4\sqrt{2} + 1 = 9 - 4\sqrt{2}$

$(1 - \sqrt{2})^2 = 1^2 - 2 \cdot 1 \cdot \sqrt{2} + (\sqrt{2})^2 = 1 - 2\sqrt{2} + 2 = 3 - 2\sqrt{2}$

Adding these results:

$ (9 - 4\sqrt{2}) + (3 - 2\sqrt{2}) = 12 - 6\sqrt{2} $

This simplifies to $ 6(2 - \sqrt{2}) $.

Thus, the sum of the squares of the roots is $ 6(2 - \sqrt{2}) $.

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The sum, of the squares of all the roots of the equation $x^2+|2 x-3|-4=0$, is