Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

The next set of medium/hard PS algebra questions. I'll post OA's with detailed explanations after some discussion. Please, post your solutions along with the answers.

1. If \(x=\sqrt[4]{x^3+6x^2}\), then the sum of all possible solutions for x is:

1. If \(x=\sqrt[4]{x^3+6x^2}\), then the sum of all possible solutions for x is:

A. -2 B. 0 C. 1 D. 3 E. 5

Take the given expression to the 4th power: \(x^4=x^3+6x^2\);

Re-arrange and factor out x^2: \(x^2(x^2-x-6)=0\);

Factorize: \(x^2(x-3)(x+2)=0\);

So, the roots are \(x=0\), \(x=3\) and \(x=-2\). But \(x\) cannot be negative as it equals to the even (4th) root of some expression (\(\sqrt{expression}\geq{0}\)), thus only two solution are valid \(x=0\) and \(x=3\).

The sum of all possible solutions for x is 0+3=3.

Answer: D.

Hi Bunuel,

Query here.

First of all we don't necessarily have to solve for all the three roots. Since, it is asking for a composite value i.e. the sum, we can directly obtain it.

We can rearrange to get:-

\(x^4-x^3-6x^2=0\) Over here, by the system of poly equations Sum of roots = - (\(coeff ofx^3/coeff of x^4\)) Similarly, Sum of roots taken two at a time = (\(coeff ofx^2/coeff of x^4\)) Product of roots = "Zero" as there is no constant term.

Also, -2 satisfies the equation \(x^4-x^3-6x^2=0\)

So, shouldn't the answer be -1, with roots as 0,-2,3?

It's explained in the solution above: \(x\) cannot be negative as it equals to the even (4th) root of some expression (\(\sqrt{expression}\geq{0}\)), thus only two solution are valid \(x=0\) and \(x=3\).
_________________

1. If \(x=\sqrt[4]{x^3+6x^2}\), then the sum of all possible solutions for x is:

A. -2 B. 0 C. 1 D. 3 E. 5

Kudos points for each correct solution!!![/quote]

X^4 = X^3 + 6x^2.Solving this we get teh values of x as 0, 3 , -2. But X cannot be negative as X is the positive fourth root of an expression(\sqrt{Expression}>= 0) So X = 0 and 3 Sum of the values = 0+3 = 3

2. The equation x^2 + ax - b = 0 has equal roots, and one of the roots of the equation x^2 + ax + 15 = 0 is 3. What is the value of b?

A. -64 B. -16 C. -15 D. -1/16 E. -1/64 Kudos points for each correct solution!!![/quote]

Since 3 is the root of the equation x^2 + ax + 15 = 0, therefore 3 must satisfy the equation x^2 + ax + 15 = 0 . On putting x=3 in the equation x^2 + ax + 15 = 0 , we get a= -8 On putting the value of a in the original equation x^2 + ax - b = 0, we get x^2 - 8x - b = 0. As per question x^2 - 8x - b = 0 has equal roots so Discriminant(D) = b^2 -4ac = 0 = (-8)^2 - 4.1.(-b)= 0. From this we get the value of b as -16

3. If a and b are positive numbers, such that a^2 + b^2 = m and a^2 - b^2 = n, then ab in terms of m and n equals to:

A. \(\frac{\sqrt{m-n}}{2}\) B. \(\frac{\sqrt{mn}}{2}\) C. \(\frac{\sqrt{m^2-n^2}}{2}\) D. \(\frac{\sqrt{n^2-m^2}}{2}\) E. \(\frac{\sqrt{m^2+n^2}}{2}\) Kudos points for each correct solution!!![/quote]

Adding the given two equations we get 2a^2 = (m+n)/2 Subtracting the given two equations we get 2b^2 = (m-n)/2

Multiplying above two equations we get 4a^2b^2 = (m^2 - n^2) => ab = \(\frac{\sqrt{m^2-n^2}}{2}\)

4. What is the maximum value of -3x^2 + 12x -2y^2 - 12y - 39 ?

A. -39 B. -9 C. 0 D. 9 E. 39 Kudos points for each correct solution!!![/quote]

Rearranging the given equation we get -3(x^2 - 4x + 4) -2(y^2 + 6y +9) - 9 = -3(x -2)^2 - 2(y +3)^2 -9 So, we need to maximize the value of -3(x -2)^2 - 2(y +3)^2 -9

Since, the maximum value of -3(x -2)^2 and - 2(y +3)^2 is zero when x= 2 and y = -3, then the maximum value of the whole expression is 0 + 0 -9 = -9

5. If x^2 + 2x -15 = -m, where x is an integer from -10 and 10, inclusive, what is the probability that m is greater than zero?

A. 2/7 B. 1/3 C. 7/20 D. 2/5 E. 3/7 Kudos points for each correct solution!!![/quote]

On Re-arranging the given equation we get m = - x^2 - 2x +15

For m to be positive - x^2 - 2x +15 should be greater then 0. Solving we get (x + 5)(x - 3)<0. Since x is an integer we get the values of x as -4 , -3, -2, -1, 0, 1, 2 (total 7 values)

Given that X is an integer from -10 and 10, inclusive (21 values)

7. If x^4 = 29x^2 - 100, then which of the following is NOT a product of three possible values of x?

I. -50 II. 25 III. 50

A. I only B. II only C. III only D. I and II only E. I and III only Kudos points for each correct solution!!![/quote]

Rearranging the equation we get x^4 - 29x^2 + 100 = 0 => (x^2 - 25)(x^ - 4) =0 => x= +5, -5, +2, -2 From these values of x we can get -50 (-5 * +5 * +2), +50(-5 * +5 * -2) but we can't get 25

8. If m is a negative integer and m^3 + 380 = 381m , then what is the value of m?

A. -21 B. -20 C. -19 D. -1 E. None of the above

on re-arranging we get m^3 -381m = -380 => m(m^2 - 381)= 380 On checking the options and hit and trial, we get the values of m to be 1 and -20. But since m is negative so m = -20

1. If x=\sqrt[4]{x^3+6x^2}, then the sum of all possible solutions for x is:

A. -2 B. 0 C. 1 D. 3 E. 5

Solution: new-algebra-set-149349-60.html#p1200948

x^4=x^3 + 6x^2 x^4 - x^3 - 6x^2 = 0 ------------> x^2 (x^2 - x - 6) = 0 -----------> x^2 (x-3)(x+2) = 0 x=0 or 3 or -2. With x = -2 original equation does not hold true so possible values for X are 3 and 0 Hence their sum is 3 Choice D.

2. The equation x^2 + ax - b = 0 has equal roots, and one of the roots of the equation x^2 + ax + 15 = 0 is 3. What is the value of b?

A. -64 B. -16 C. -15 D. -1/16 E. -1/64

Solution: new-algebra-set-149349-60.html#p1200950

x^2 + ax - b = 0 that means a^2 - 4(-b) = 0 ------> a^2 + 4b = 0 -------------Equation I x^2 + ax + 15 has 3 as a root, so 9 + 3a + 15 = 0 -------> 3a + 24 = 0 --------> a = -8 We will put the value of a in equation I -----------> a^2 + 4b = 0 --------> 64 + 4b = 0 ------> b = -16 ----> Choice B is the answer.
_________________

5. If x^2 + 2x -15 = -m, where x is an integer from -10 and 10, inclusive, what is the probability that m is greater than zero?

A. 2/7 B. 1/3 C. 7/20 D. 2/5 E. 3/7

Solution: new-algebra-set-149349-60.html#p1200970

there are 21 possible values of x so does m has 21 possible values from x = -4 to x = 2, m gives negative value; Total 7 values so the probability m being negative is 7/21----> 1/3 ----> choice B

6. If mn does not equal to zero, and m^2n^2 + mn = 12, then m could be:

I. -4/n II. 2/n III. 3/n

A. I only B. II only C. III only D. I and II only E. I and III only

Solution: new-algebra-set-149349-60.html#p1200973

m^2n^2 + mn = 12 ------> mn(mn+1) = 12 ----------> here mn and mn+1 are consecutive integers. So 12 is basically a product of 2 consecutive intergers those integers must be 3 and 4 or -3 and -4 so possible values for mn are mn = 3 and mn+1 = 4 --------> m = (3/n) mn = -4 and mn + 1 =-3 -------> m =-(4/n) Choice E
_________________

7. If x^4 = 29x^2 - 100, then which of the following is NOT a product of three possible values of x?

I. -50 II. 25 III. 50

A. I only B. II only C. III only D. I and II only E. I and III only

Solution: new-algebra-set-149349-60.html#p1200975

x^4 - 29x^2 + 100 = 0 --------> let X^2 be y -------> y^2 - 29y + 100 = 0 ------> y^2 - 25y - 4y + 100 = 0 --------> (y-25)(y-4)=0 y=25 -----> x^2 = 25 -----> x= 5 or -5 y=4-------> x^2 = 4 ------> x = 2 or -2

Possible values of x are 5, -5, 2, and -2 Out of options 25 can never be a product of any 3 values of x, so Choice B is the answer

8. If m is a negative integer and m^3 + 380 = 381m , then what is the value of m?

A. -21 B. -20 C. -19 D. -1 E. None of the above

Solution: new-algebra-set-149349-60.html#p1200980

This was a bit complex one m^3 - 381m = - 380 ------> m(m^2 - 381) = - 380 Since -380 is the product of m(m^2 - 381), any one of either m or (m^2 - 381) must be negative. Since we are given that m is a negative integer, then m^2 - 381 must be positive So m^2 - 381 > 0 -------> m^2 > 381 ---------> |m| > 19.5 ----approx. so m > 19.5-------if m is positive and m < 19.5 ------ if m is negative We know that m is negative so m must equal to -20 : choice B
_________________

1. If \(x=\sqrt[4]{x^3+6x^2}\), then the sum of all possible solutions for x is:

A. -2 B. 0 C. 1 D. 3 E. 5

Take the given expression to the 4th power: \(x^4=x^3+6x^2\);

Re-arrange and factor out x^2: \(x^2(x^2-x-6)=0\);

Factorize: \(x^2(x-3)(x+2)=0\);

So, the roots are \(x=0\), \(x=3\) and \(x=-2\). But \(x\) cannot be negative as it equals to the even (4th) root of some expression (\(\sqrt{expression}\geq{0}\)), thus only two solution are valid \(x=0\) and \(x=3\).

The sum of all possible solutions for x is 0+3=3.

Answer: D.

Hi Bunuel,

Could you please explain why -2 is not a solution? Is it because x is a non-negative number? Because if I plug in the value of -2 to the equation \(x^4=x^3+6x^2\) I get \(-2=\sqrt[4]{16}\) which can be true isn't it, as \(\sqrt[4]{16}=|2|\)

Also could you explain why even root of an expression cannot be negative when we have an expression which is a square, because we know that \(\sqrt{{x^2}}=|x|\)?

ohhhhhhhhhhhhhhhhhhhhh I thought that m is raised to the power 2n and the whole bracket raised to the power 2 ...that is what made me confused . Thanks a million , Bunuel
_________________

1. If \(x=\sqrt[4]{x^3+6x^2}\), then the sum of all possible solutions for x is:

A. -2 B. 0 C. 1 D. 3 E. 5

Take the given expression to the 4th power: \(x^4=x^3+6x^2\);

Re-arrange and factor out x^2: \(x^2(x^2-x-6)=0\);

Factorize: \(x^2(x-3)(x+2)=0\);

So, the roots are \(x=0\), \(x=3\) and \(x=-2\). But \(x\) cannot be negative as it equals to the even (4th) root of some expression (\(\sqrt{expression}\geq{0}\)), thus only two solution are valid \(x=0\) and \(x=3\).

The sum of all possible solutions for x is 0+3=3.

Answer: D.

Hi Bunuel,

Could you please explain why -2 is not a solution? Is it because x is a non-negative number? Because if I plug in the value of -2 to the equation \(x^4=x^3+6x^2\) I get \(-2=\sqrt[4]{16}\) which can be true isn't it, as \(\sqrt[4]{16}=|2|\)

Also could you explain why even root of an expression cannot be negative when we have an expression which is a square, because we know that \(\sqrt{{x^2}}=|x|\)?

Thank you

Note the Important Difference.

On the GMAT if X^2 = 4 then x = +/- 2 or |x| = 2 However if x= sq.root(4) then x has to be positive i.e. 2 and it can not take negative value. when we plug -2 as the value of x in the equation we would get -2 = 4th root of 16 -----> -2 = 2 This is because fourth root of 16 is 2 and not -2

The rule is even root of a number can not be negative on the GMAT

Regards,

Abhijit.

Hey Abhijit,

This is a very interesting point that you have made here. This statement that -2 is the fourth root of 16 is admissible under normal circumstances but is not in a GMAT question. Can you mention a source which would list out such peculiar rules for GMAT quants?

Thanks in advance

Anshuman
_________________

When you feel like giving up, remember why you held on for so long in the first place.

gmatclubot

Re: New Algebra Set!!!
[#permalink]
12 Apr 2013, 04:25

It’s quickly approaching two years since I last wrote anything on this blog. A lot has happened since then. When I last posted, I had just gotten back from...

Since my last post, I’ve got the interview decisions for the other two business schools I applied to: Denied by Wharton and Invited to Interview with Stanford. It all...

Marketing is one of those functions, that if done successfully, requires a little bit of everything. In other words, it is highly cross-functional and requires a lot of different...