1. ## Inequalities

Hi! Please, help to solve the inequality :
1) -х²+3x-5>0 ;
2) 3х²-4x+8≥0 .

Tnk U!

2. ## Re: Solving inequalities

Can you solve -x^2 + 3x - 5 = 0 ?

3. ## Re: Solving inequalities

Originally Posted by AnnaPorter
Hi! Please, help to solve the inequality :
1) -х²+3x-5>0 ;
2) 3х²-4x+8≥0 .
@Dan It would best if this were moved to a new thread.

1) Now that has no real roots bec, $(3)^2-4(-1)(-5)<0$ SEE HERE
BUT $x=1$ we have $-(1)^2+3(1)-5<0 so there is non solution. WHY? 2) The discriminate here is$(-4)^2-4(3)(8)<0 $again there are no real roots SEE HERE but this for$x=1$we have 3(1)^2-4(1)+8>0$ so that means it is true everywhere. WHY?

4. ## Re: Inequalities

Another way of looking at these:
Completing the square
$\displaystyle -x^2+ 3x- 5= -(x^2- 3x+ (9/4- 9/4))- 5= -(x^2- 3x+ 9/4)+ 9/4- 5$
$\displaystyle -(x- 3/2)^2+ 9/4- 20/4= -(x- 3/2)^2- 11/4$
We can think of this as -11/4 minus a square so it is never larger than -11/4 so never larger than 0.

$\displaystyle 3x^2- 4x+ 8= 3(x^2- (4/3)x+ (4/9- 4/9))+ 8= 3(x^2- (4/3)x+ 4/9)- 4/3+ 8$
$\displaystyle 3(x+ 2/3)^2- 4/3+ 24/3= 3(x+2/3)^2+ 5$

That is 5 plus a square to is never less than 5 and so never less than 0.