Which value(s) of x satisfies the equation above?

Question

2x - 2 = \sqrt{3x^2+13} Which value(s) of x satisfies the equation above? I. -1 II. 4 III. 9 (A) I (B) III (C) I & II (D) I & III (E) I, II, & III For a discussion of algebraic equations involving radicals, as well as a solution to this question, see this post: https://magoosh.com/gmat/2013/gmat-math ... -radicals/ Mike

Zarrolou · Accepted Answer

Using numbers:

2x - 2 = \sqrt{3x^2+13}

I)-1

-4= \sqrt{3(-1)^2+13}

\sqrt{3+13}  does not equal  -4  , so  -1  is NOT a possible value.

If we take a look at the possible answer, all contain I except B. So B is the correct answer

PareshGmat · Answer

I solved the equation to get the wrong answer. Squaring both sides made the difference.

Leason Learnt: Such type of questions, better to place the given values & check.

Bunuel · Answer

Similar question to practice: new-algebra-set-149349-60.html#p1200948

Ashishmathew01081987 · Answer

2x-2 = \sqrt{3x^2 + 13}

Squaring both sides, we get

4x^2 - 4x + 3 = 3x^2 + 13

x^2 - 4x = 9

x (x-4) = 9

x = 9 or x-4 = 9

x = 9 or x = 13

Since only X = 9 is given and it satisfies the condition 
 Answer is B

sreelakshmigs · Answer

Hey Ashishmathew,
(2x-2)^2 = 4x^2 +4-8x right ? How come ur LHS of the eqn is   4x^2 - 4x + 3 ? Or Am I missing out something here ?

mikemcgarry · Answer

Dear sreelakshmigs I'm happy to respond. :-)

First of all, Ashishmathew01081987 's solution is not correct at all. You are perfectly correct: (2x-2)^2 = 4x^2 - 8x + 4 Also, the factoring thing he does at the end, the steps after x(x - 4) = 9, are 100% incorrect. If you want to see the correct solution to this problem, see: https://magoosh.com/gmat/2013/gmat-math- ... -radicals/ Mike :-)

Ashishmathew01081987 · Answer

Hey Ashishmathew, (2x-2)^2 = 4x^2 +4-8x right ? How come ur LHS of the eqn is 4x^2 - 4x + 3 ? Or Am I missing out something here ? Dear sreelakshmigs I'm happy to respond. :-)

First of all, Ashishmathew01081987 's solution is not correct at all. You are perfectly correct: (2x-2)^2 = 4x^2 - 8x + 4 Also, the factoring thing he does at the end, the steps after x(x - 4) = 9, are 100% incorrect. If you want to see the correct solution to this problem, see: https://magoosh.com/gmat/2013/gmat-math- ... -radicals/ Mike :-)

Thanks Sree for pointing out my mistake. That was a blunder. Hope that it doesn't happen on the GMAT.

Thanks Mike, I understand why the factoring stuff would have led me into the trap. Plugging numbers is the best option here.

mikemcgarry · Answer

Dear Ashishmathew01081987 , Actually, it's very good to understand the algebra in this problem. Again, you can see a full algebraic solution at the blog to which I linked. Plugging numbers is good sometimes, but it's best not to make that a one-size-fit-all kind of strategy. Does this make sense? Mike :-)

gbatistoti · Answer

I think this might be incorrect.

First, -4 = 16^(1/2) is as correct answer since (-4)(-4) = 16.
Second, the bold part is not correct. x² when x = -1 is the same as -1² = -1 and not (-1)² = 1, so we should have -4 = 10^(1/2). So, option I is incorrect.
Since the only option that says I is incorrect is B, then, B is our answer.

BrentGMATPrepNow · Answer

We can solve this question in a matter of seconds if we recognize that we aren't required to actually solve the given equation. 
Instead, we can test each answer choice.

I. -1 
Is x = -1 a solution to the equation? 
Let's plug it in and find out:  2(-1) - 2 = \sqrt{3(-1)^2+13} 
STOP!
The left side of the equation evaluates to be -4
Since the square root of a number cannot be negative, we know that x = -1 is NOT a solution.

This means we can eliminate answer choices A, C, D and E.

By the process of elimination, the correct answer is B

abhinav261289 · Answer

Mathematically , both (-4)^2 and (4)^2 can give 16 . So (16)^1/2 could be -4, and -1 could also be a root. Could someone please help me understand that what am I missing here ?

MathRevolution · Answer

Rather than squaring both sides to get the value of 'x, it is much easier to substitute the options.

A simple simplification:  (2x -2)^2 = 3x^2  + 13

I: -1: 2x - 2 = 2(-1) - 2 = -4 =  (-4)^2  = 16 =>  3x^2 + 13 = 3(-2)^2 + 13  - more than 16 - NOT POSSIBLE.

II: 4: 2x - 2 = 2(4) - 2 = 6 =  (6)^2 = 36 => 3x^2 + 13 = 3(6)^2  + 13 - more than 36 - NOT POSSIBLE.

III: 9: 2x - 2 = 2(9) - 2 = 16 =  (16)^2 = 256 => 3x^2 + 13 = 3(9)^2  + 13 = 243 + 13 = 256 - POSSIBLE.