M31-19

Question

What is the range of all the roots of  |x^2 - 2| = x ?

A. 4
B. 3
C. 2
D. 1
E. 0

Bunuel · Accepted Answer

Official Solution:

What is the range of all the roots of  |x^2 - 2| = x ?

A. 4
B. 3
C. 2
D. 1
E. 0

First, note that since  x  is equal to the absolute value of some number ( |x^2 - 2| ),  x  cannot be negative.
 
Next,  |x^2 - 2| = x  implies that either  x^2 - 2 = x  or  -(x^2 - 2) = x . 
 
The first equation,  x^2 - 2 = x , yields  x = -1  or  x = 2 . Since  x  cannot be negative, we only consider  x = 2 .
 
The second equation,  -(x^2 - 2) = x , yields  x = -2  or  x = 1 . Again, since  x  cannot be negative, we only consider  x = 1 .
 
The range is the difference between the largest and smallest values: range = 2 - 1 = 1.

Answer: D

monika25 · Answer

I think this is a  high-quality  question.

Radheya · Answer

Hello  Bunuel 
For this particular question, can we not square both the sides?

|x^2 - 2| = x 
(x^2 - 2)^2 = x^2 (squaring both the sides)
x^2 + 4 -4x = x^2 (expanding the equations)
4x = 4 (Subtracting x^2)
x = 1
and hence, D?

Though I completely agree and understood your above explanation and I also remember from one of you post that as we don't know whether the mod value is positive or negative we should not square and look for an alternative way to solve the question, can you elaborate on the difference between the two approaches? 
Thanks.

Bunuel · Answer

1. We can square equations if both sides are non-negative (for example, if it were |x^2 - 2| = |x|, then squaring would be correct). When that's not the case squaring usually creates more roots, then there actually are.

2. Highlighted part is not correct:  (x^2 - 2)^2=x^4 - 4 x^2 + 4 , notice that x on the left hand side is in fourth power not squared.

3. If you solve  x^4 - 4 x^2 + 4=x^2 , you'll get that x can be -2, -2, 1, or 2. But -2 and -1 do not satisfy |x^2 - 2| = x, and should be discarded. So, we are left with only x = 1 and x = 2. As you can see, squaring gave more roots, then there actually are.

Hope it helps.

Alok322 · Answer

VeritasKarishma ,

Can we solve this question using graphs that you discussed in one of the posts? I tried using that method but the only relevant data that I could get was that we can have 2 roots. But I could not find what the roots are from the graphs. Am I missing something?

KarishmaB · Answer

The easiest way to deal with this question is to square and find out which roots satisfy the original equation. Graphing helps you see that there are two roots. One will be between 0 and sqrt(2) and the other a bit greater than sqrt(2). Here, I could see that 1 and 2 satisfy the equation though some quadratic solving would be required if you were unable to guess the roots. Finding the roots is easier with lines because of the linear relation between x and y.

RenanBragion · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

MyNameisFritz · Answer

Fantastic question! It's key not only to answer what is asked but also to remember that the roots can not be negative in this case.

pyreddy · Answer

hi sir . here why we are not taking negatives inti the consideration. ?

Bunuel · Answer

We don’t take negative x because the equation is |x^2 - 2| = x.

The left side is always non-negative because of the absolute value. So the right side, x, must also be non-negative.

If x were negative, the two sides couldn’t be equal.

mbaaccount1234 · Answer

Bunuel  shouldnt the range be 2 -(-1) = 3?

Range = highest - lowest

Bunuel · Answer

The solutions for |x^2 - 2| = x are only x = 2 and x = 1. x = -1 does not satisfy the equation.

So the range = highest - lowest = 2 - 1 = 1.

Manvi01 · Answer

I like the solution - it’s helpful.

Unknown5612 · Answer

Bunuel

Case 1: x^2>=2: (x^2)-2=x, solve,will get x=2,-1...since x^2>=2, only x=2 satisfies
Case 2: x^2<2: (x^2)-2=-x, solve, will get x=-2,1, since x^2<2, only x=1 satisfies
So range=2-1=1

Is this method correct?

agrasan · Answer

I like the solution - it’s helpful.

RV64 · Answer

I like the solution - it’s helpful.

luisdicampo · Answer

Deconstructing the Question  
We are given the equation:
 |x^2 - 2| = x 
and asked to find the range of all real roots.

Step-by-step  
Since the left side is an absolute value, it is always nonnegative, so:
 x ≥ 0

Case 1:  x^2 - 2 ≥ 0  
 |x^2 - 2| = x^2 - 2 
 x^2 - 2 = x 
 x^2 - x - 2 = 0 
 (x - 2)(x + 1) = 0 
Valid solution:  x = 2

Case 2:  x^2 - 2 < 0  
 |x^2 - 2| = 2 - x^2 
 2 - x^2 = x 
 x^2 + x - 2 = 0 
 (x + 2)(x - 1) = 0 
Valid solution:  x = 1

The solution set is:
 \{1, 2\}

Range:
 2 - 1 = 1

Answer: 1

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