Which of the following is always equal to √(9 + x^2 -6x) ?

Question

Which of the following is always equal to √(9 + x^2 -6x) ?
A.x -3
B.3 + x
C.|3 – x|
D.|3 +x|
E.3 – x

OA is C

Please help on this one. I don’t feel this question is correct because after factorization it gives
√(x-3)^2 that is equal to |x – 3| i.e √(x-3)^2 = |x – 3|

So, what is X here positive or negative.

theCodeToGMAT · Answer

√(9 + x^2 -6x) it can be sqrt(3-x)^2 = 3-x or sqrt(x-3)^2 = x-3 Since, |3-x| is either 3-x or x-3 {C}

Uva@90 · Answer

vinni.k wrote:Which of the following is always equal to √(9 + x^2 -6x) ?
A.x -3
B.3 + x
C.|3 – x|
D.|3 +x|
E.3 – x

OA is C

Let x=1.
In this case, √(9 + x² - 6x) = √4 = 2.
Now plug x=1 into the answer choices and eliminate any that don't yield a value of 2.

A. x -3 = 1-3 = -2.
B. 3 + x = 3 + 1 = 4.
C. |3 – x| = |3-1| = 2.
D. |3 +x| = |3+1| = 4.
E. 3 – x = 3-1 = 2.
Eliminate A, B and D.

Test an EXTREME value.
Let x=10.
In this case, √(9 + x² - 6x) = √49 = 7.
Now plug x=10 into the remaining answer choices and eliminate any that don't yield a value of 7.

C. |3 – x| = |3-10| = 7.
E. 3 – x = 3-10 = -7.
Eliminate E.

The correct answer is C.

vinni.k · Answer

Thanks Rahul. I am not sure how to follow you. How you got this one sqrt(3-x)^2 = 3-x ? This is fine:- sqrt(x-3)^2 = x-3 but not sure about the above one. Vinni

theCodeToGMAT · Answer

vinni.k wrote:Thanks Rahul.
I am not sure how to follow you. How you got this one sqrt(3-x)^2 = 3-x ?
This is fine:- sqrt(x-3)^2 = x-3 but not sure about the above one.
Vinni

This is because (3-x)^2 = 9 + x^2 - 6x
and
(x-3)^2 = x^2 + 9 - 6x
both blue parts are same

vinni.k · Answer

Uva@90 wrote: What you did is correct.
I assume you have confused with |x-3| and |3-x|
Both yield same values.

You ended as √(x-3)^2 = |x – 3| which is = |3-x|
hence answer is C

Regards,
Uva.

Thanks Uva, but can you please prove how |x - 3| is |3 - x|. I am just confused between A and C.

Vinni

Uva@90 · Answer

vinni.k wrote:Uva@90 wrote: What you did is correct. I assume you have confused with |x-3| and |3-x| Both yield same values. You ended as √(x-3)^2 = |x – 3| which is = |3-x| hence answer is C Regards, Uva. Thanks Uva, but can you please prove how |x - 3| is |3 - x|. I am just confused between A and C. Vinni Hi Vinni, Let us consider X = 1 √(9 + x^2 -6x) =2 and |x - 3| =2 and |3 - x| is also 2 Let us consider X = -1 √(9 + x^2 -6x) =4 |x - 3| =4 and |3 - x| is also 4 I am just confused between A and C. Let us consider simple one, √x^2 = x when x>0 and -x when x <0 similarly, √(9 + x^2 -6x) should have both values (x-3) and (-x+3) Option A has only one value. Where as C has both. Hence Ans is C Regards, Uva.

vinni.k · Answer

Okay. I think i got your point. Clearly understandable. Thank you and really appreciate it. Regards Vinni

GMATGuruNY · Answer

vinni.k wrote:Which of the following is always equal to √(9 + x^2 -6x) ?
A.x -3
B.3 + x
C.|3 – x|
D.|3 +x|
E.3 – x

OA is C

Let x=1.
In this case, √(9 + x² - 6x) = √4 = 2.
Now plug x=1 into the answer choices and eliminate any that don't yield a value of 2.

A. x -3 = 1-3 = -2.
B. 3 + x = 3 + 1 = 4.
C. |3 – x| = |3-1| = 2.
D. |3 +x| = |3+1| = 4.
E. 3 – x = 3-1 = 2.
Eliminate A, B and D.

Test an EXTREME value.
Let x=10.
In this case, √(9 + x² - 6x) = √49 = 7.
Now plug x=10 into the remaining answer choices and eliminate any that don't yield a value of 7.

C. |3 – x| = |3-10| = 7.
E. 3 – x = 3-10 = -7.
Eliminate E.

The correct answer is C.

Matt@VeritasPrep · Answer

One last (succinct) way: √z² = |z| for any value of z. So ... √(x² - 6x + 9) = √(x-3)(x-3) = √(x-3)² = |x-3|. Hence the answer is C!

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