What is the product of the roots of the equation y = |x|^2 - 5|x| + 6

Question

What is the product of the roots of the equation  y = |x|^2 - 5|x| + 6 ?

(A) -36
(B) 6
(C) -6
(D) 36
(E) 0

Are You Up For the Challenge: 700 Level Questions

Find the product of the roots of the equation  y = |x|^2 - 5|x| + 6 , where |x| is the modulus of x.

pardhu1212 · Accepted Answer

When the equation is in the form of  y=ax^2+bx+c , the product of the roots of the equation is  \frac{c}{a} .

The equation in the question can be split into two different equations  x^2-5x+6  and  x^2+5x+6  when  x>6  and  x<6  respectively.

The final answer is the product of the products of roots of these two equations. In each equation, the product is  6/1=6 . Hence the final answer is 36.

gvij2017 · Answer

|X| = 3, 2
X= -3,3,-2 and 2
Product of all values of X = 36
 D is correct.

ultimateashish · Answer

Trick is to realize that |x|^2 will always be positive irrespective of the sign of x.
If x<0 then |x|=-x but |x|^2=(-x)^2=x^2
If x>=0 then |x|=x and |x|^2=x^2

So the equation can be converted to y=x^2 -5|x| +6
Then solve for the roots as 3,2 and -3,-2
Product of roots = 36

Abhishek009 · Answer

Treat  y = |x|^2 - 5|x| + 6  as  y = x^2 - 5x + 6

Or,  y = x^2 -3x - 2x + 6

Or,  y = x(x -3) - 2( x - 3)

Or,  y = (x -3)(x - 2)

Now if  y = 0  we have  x = 3 , 2

Now, initially we have assumed only the +ve values of  y , as  3 , 2 , so the other possible values are  -3 , -2

Thus, the product will be  3*2*-3*-2 = 36 , Hence Answer must be (D)

BrushMyQuant · Answer

We need to find the product of the roots of the equation  y = |x|^2 - 5|x| + 6 ?

y = |x|^2 - 5|x| + 6  
=>  y = x^2 - 5|x| + 6  (|x|^2 is same as x^2, both are non-negative)

As we have |x| in the equation so we will have two cases 
 
   -   Case 1:  x ≥ 0
 
=> |x| = x
=>  x^2 – 5x + 6 = 0 
=>  x^2 -2x -3x + 6 = 0 
=> x*(x - 2) - 3*(x - 2) = 0
=> (x - 2) * (x - 3) = 0
=> x = 2, 3

But condition was x ≥ 0 and both 2 and 3 are ≥ 0 
=>  x = 2, 3 are  SOLUTIONS   
  -   Case 2:  x ≤ 0
 
=> |x| = -x
=>   x^2 + 5x + 6 = 0  
 =>  x^2 + 2x + 3x + 6 = 0 
=> x*(x + 2) + 3*(x + 2) = 0
=> (x + 2) * (x + 3) = 0
=> x = -2, -3

But condition was x ≤ 0 and both -2 and -3 are ≤ 0 
=>  x = -2, -3 are  SOLUTIONS

So, Product of the roots = 2 * 3 * -2 * -3 = 36

So,  Answer will be D 
Hope it helps!

Watch the following video to learn the Basics of Absolute Values

https://www.youtube.com/watch?v=huFiBk8PaGY

Vibhatu · Answer

This is the easiest. 
Product of the roots, c/a
In this case 6/1 and 6/1 
  answer is 36: D

saurvan · Answer

I solved this by using P.O.R=C/A, c=36, since |x|^2 is given, a is always 1 , so POR=36. Am I wrong here?

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

What is the product of the roots of the equation y = |x|^2 - 5|x| + 6

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

As we have \|x\| in the equation so we will have two cases
-Case 1: x ≥ 0 => \|x\| = x => \(x^2 – 5x + 6 = 0\) => \(x^2 -2x -3x + 6 = 0\) => x(x - 2) - 3(x - 2) = 0 => (x - 2) * (x - 3) = 0 => x = 2, 3 But condition was x ≥ 0 and both 2 and 3 are ≥ 0 => x = 2, 3 are SOLUTIONS	-Case 2: x ≤ 0 => \|x\| = -x => \( x^2 + 5x + 6 = 0 \) => \(x^2 + 2x + 3x + 6 = 0\) => x(x + 2) + 3(x + 2) = 0 => (x + 2) * (x + 3) = 0 => x = -2, -3 But condition was x ≤ 0 and both -2 and -3 are ≤ 0 => x = -2, -3 are SOLUTIONS

GMAT Problem Solving (PS) Questions

What is the product of the roots of the equation y = |x|^2 - 5|x| + 6

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards