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?

SmileAndSolve · Answer

The equation is:

y=∣x∣
2
−5∣x∣+6

But the question asks for the roots.

What does "roots" mean?

Roots are the values of x where:

y=0
 ​

So we set y=0 — not because y is assumed negative, but because that's what a root means.

Thus:

0=∣x∣
2
−5∣x∣+6

Let a=∣x∣:

a
2
−5a+6=0

Factor:

(a−2)(a−3)=0

So:

∣x∣=2or∣x∣=3

Therefore:

x=2,−2,3,−3

RaahiK · Answer

Vieta's formula (c/a) applies only to polynomial equations in x. Here the equation contains ∣x∣, so it is not a polynomial.
Split into cases: x≥0 gives roots 2,3
 x<0 gives roots −2,−3.
Hence the product of all roots =36

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

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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

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