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

Question

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

Accepted Answer

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

Case I: $x\ge \frac{3}{2}$
For $x\ge \frac{3}{2}$, 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:

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

Case II: $x
For $x${x}^{2}+3-2x-4=0$

Simplifying gives:

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

Solving this quadratic equation, we obtain:

$x=1-\sqrt{2}$

Sum of the Squares of the Roots
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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