How do you define this quadratic absolute function?

[brandoncasilla]

How do you define this |x^2x+2|=3x-4

defining where its like y={(equation) if x<(a number) or x>(a number)
-(equation) if (a number)< X < (anumber)

 

[pka]

How do you define this |x^2x+2|=3x-4 defining where its like y={(equation) if x<(a number) or x>(a number)-(equation) if (a number)< X < (anumber)

Is it \(\displaystyle x^{2x}+2\) or is it \(\displaystyle x^2+x+2~?\)

 

[brandoncasilla]

its the 2nd one

 

[pka]

its the 2nd one

Then solve \(\displaystyle (x^2+x+2)^2=(3x-4)^2\). Choose the solution that makes \(\displaystyle 3x-4\ge 0\).

That is just square both sides: \(\displaystyle |a|=|b|\text{ if and only if }a^2=b^2~.\)

 

[HallsofIvy]

What you should know if you are asked a question like this is "|A|= A as long as \(\displaystyle A\ge 0\), |A|= -A if A< 0". Here "A" is \(\displaystyle x^2+ x+ 2\) so "\(\displaystyle |x^2+ x+ 2|= x^2+ x+ 2= 3x+ 4\) as long as \(\displaystyle x^2+ x+ 2\ge 0\) and \(\displaystyle |x^2+ x+ 2|= -(x^2+ x+ 2)= 3x+ 4\) if \(\displaystyle x^2+ x+ 2< 0\)".

Can you solve \(\displaystyle x^2+ x+ 2= 3x+ 4\)? Can you solve \(\displaystyle -(x^2+ x+ 2)= 3x+ 4\)? Can you determine where \(\displaystyle x^2+ x+ 2\) is positive and where it is negative?