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[Solved] Absolute Value Equations

 
Linear Equations
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June 14, 2021 5:57 pm
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(@armor)
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Given a real number d and   |d|\leq \frac{1}{4},  solve the equation   x^4-2x^3+(2d-1)x^2+2(1-d)x+2d+d^2=0.

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June 14, 2021 5:57 pm
Topic starter
Staff 
(@armor)
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Rewrite the equation as  \small d^2+(2x^2-2x+2)d+x^4-2x^3-x^2+2x=0  and treat it a quadratic equation for d, then the quadratic formula implies  \small d=-x^2-x   or  \small d=-x^2+3x-2.  Both are quadratic equations for x. Solve them to obtain four roots of the original equation: 

\small x=\frac{-1+\sqrt{1-4d}}{2}, \frac{-1-\sqrt{1-4d}}{2},\frac{3+\sqrt{4d+1}}{2}, \frac{3-\sqrt{4d+1}}{2}.

All these roots exist since  \small |d|\leq \frac{1}{4}.

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