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Algebra Level 3

Consider the following:

• Let ${ S }_{ 1 }$ be the sum of the roots of the equation $4{ x }^{ 2 }+5-8x=0$

• Let ${ S }_{ 2 }$ be the sum of the roots of the equation $\log _{ 6 }{ \left( x-2 \right) } +\log _{ 6 }{ \left( x+3 \right) } =2$

• Let ${ S }_{ 3 }$ be the sum of the roots of the equation $x-\cfrac { 7 }{ x-3 } =3-\cfrac { 7 }{ x-3 }$

• Let ${ S }_{ 4 }$ be the sum of the roots of the equation $4x+\sqrt { 2x } =1$

Then find the value of $Z$ where $Z={ S }_{ 1 }+{ S }_{ 2 }+{ S }_{ 3 }+{ S }_{ 4 }$.

AYWC?

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