Remember, x must be real!

Algebra Level pending

For real numbers $x$

$x$ satisfy the following equation

$1.5x+5.5-\sqrt { 33x }$ = $-|6x-22|$

If the value of $x$ can be expressed in the form $\frac { a }{ b }$ in which $a$ and $b$ are coprime positive integers, find the value of $a+b$

This problem is edited from a classic problem.

1 solution

Joshua Chin
Oct 24, 2015

Complete the square for $1.5x+5.5+\sqrt { 33x }$

$1.5x+5.5+\sqrt { 33x }$ = $\frac { 3x+2\sqrt { 33x } +11 }{ 2 }$

The above expression equals to $\frac { { (\sqrt { 3x } +\sqrt { 11 } ) }^{ 2 } }{ 2 }$

Shift $|6x-22|$ to the left hand side to get $|6x-22| +\frac { { (\sqrt { 3x } +\sqrt { 11 } ) }^{ 2 } }{ 2 }=0$

As $\frac { { (\sqrt { 3x } +\sqrt { 11 } ) }^{ 2 } }{ 2 }$ , $|6x-22|$ $\ge$ 0

This means that $\frac { { (\sqrt { 3x } +\sqrt { 11 } ) }^{ 2 } }{ 2 } =0$

By manipulating, one will get $x=\frac { 11 }{ 3 }$ .

So $a+b$ = $11+3$ = $14$