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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
_________________

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