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In the equation x^2+kxm=0 where k and m are both integers, what is m?

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Abhishekgmat87
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In the equation x^2+kxm=0 where k and m are both integers, what is m? [#permalink] New post   09 Apr 2022, 08:25
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In the equation \(x^2 + kx − m\) = 0 where k and m are both integers, what is the value of m?

(1) k = 5

(2) If both roots of the equation were negative, the value of k would be 11.
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C
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In the equation x^2+kxm=0 where k and m are both integers, what is m? [#permalink] New post   Updated on: 08 Jul 2022, 09:07
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Official Solution -

Evaluate Statement (1). Since there are variables, Plug In. If k = 5 and x = 2, then the equation in the question stem is \(2^2\) + (5)(2) – m = 0 which simplifies to 4 + 10 – m = 0 or 14 – m = 0 and m = 14. However, if k = 5 and x = 6, then the equation is 62 + (5)(6) – m = 0 which simplifies to 36 + 30 – m = 0 or 66 – m = 0 and m = 66. When different numbers that satisfy a statement yield different answers to the question, the statement is insufficient. So, write down BCE.

Now, evaluate Statement (2). It states that if both roots are negative, then k = 11. If the two roots are negative, then the two factors are positive. Since k is the sum of the last terms in the two factors and m is the product, the factors could be (x + 2)(x + 9). In that case, m = 2 × 9 =18. However, if the two factors are (x + 3)(x + 8), then m = 3 × 8 = 24. When different numbers that satisfy a statement yield different answers to the question, the statement is insufficient. Eliminate choice B.

Now, evaluate both statements together. Statement (2) states that if both roots are negative, then k = 11 and Statement (1) states that k = 5. So, in order to satisfy Statement (1), one of the roots is positive and one is negative because if both factors were positive, then the last terms would have to be negative. This means that the sum of the absolute value of the two last terms is 11 and the difference is 5. The only values that satisfy both statements are the factors (x + 8)(x – 3). Therefore, the original equation is \(x^2\) + 5x – 24 = 0 and m = 24. Since both statements together provide one specific answer to the question, which is 24, the statements together are sufficient. Eliminate choice E.

The correct answer is choice C.

Originally posted by Abhishekgmat87 on 15 Apr 2022, 20:27.
Last edited by Abhishekgmat87 on 08 Jul 2022, 09:07, edited 1 time in total.
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Re: In the equation x^2+kxm=0 where k and m are both integers, what is m? [#permalink] New post   08 Jul 2022, 03:16
The explanation posted by Abhishekgmat87 is the official one by princeton. Could anyone else help with this question. I am aware that sum of roots is -b/a and product of roots is c/a, but how does one reach option C through the information provided. A little confused.

Bunuel nick1816
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In the equation x^2+kxm=0 where k and m are both integers, what is m? [#permalink] New post   08 Jul 2022, 09:15
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gandharvm wrote:
The explanation posted by Abhishekgmat87 is the official one by princeton. Could anyone else help with this question. I am aware that sum of roots is -b/a and product of roots is c/a, but how does one reach option C through the information provided. A little confused.

Bunuel nick1816


Let the roots be a and b. m = Product of roots = - a * b
Statement 1 - Sum of roots = -k = -5
a+b = -5 (i)
We can't find m from this eqn. Not sufficient

Statement 2 - Let the roots be c and d, c+d =-11
Given both c and d are negative. We can't find value of m from this eqn. Not sufficient

Combing both - If both roots of the equation were negative, the value of k would be 11. It means both roots are not negative. Since the sum of the roots is negative (-5), one of the root is positive while the other is negative
Let us assume that a is negative and b is positive. If both are negative, sum of roots will become a-b = -11 (ii)
Solving these equations, we get a = -8 and b = 3

m = -a*b = 24 (Sufficient).
Option C
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Re: In the equation x^2+kxm=0 where k and m are both integers, what is m? [#permalink] New post   17 Jul 2022, 07:55
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Abhishekgmat87 wrote:
Official Solution -

Evaluate Statement (1). Since there are variables, Plug In. If k = 5 and x = 2, then the equation in the question stem is \(2^2\) + (5)(2) – m = 0 which simplifies to 4 + 10 – m = 0 or 14 – m = 0 and m = 14. However, if k = 5 and x = 6, then the equation is 62 + (5)(6) – m = 0 which simplifies to 36 + 30 – m = 0 or 66 – m = 0 and m = 66. When different numbers that satisfy a statement yield different answers to the question, the statement is insufficient. So, write down BCE.

Now, evaluate Statement (2). It states that if both roots are negative, then k = 11. If the two roots are negative, then the two factors are positive. Since k is the sum of the last terms in the two factors and m is the product, the factors could be (x + 2)(x + 9). In that case, m = 2 × 9 =18. However, if the two factors are (x + 3)(x + 8), then m = 3 × 8 = 24. When different numbers that satisfy a statement yield different answers to the question, the statement is insufficient. Eliminate choice B.

Now, evaluate both statements together. Statement (2) states that if both roots are negative, then k = 11 and Statement (1) states that k = 5. So, in order to satisfy Statement (1), one of the roots is positive and one is negative because if both factors were positive, then the last terms would have to be negative. This means that the sum of the absolute value of the two last terms is 11 and the difference is 5. The only values that satisfy both statements are the factors (x + 8)(x – 3). Therefore, the original equation is \(x^2\) + 5x – 24 = 0 and m = 24. Since both statements together provide one specific answer to the question, which is 24, the statements together are sufficient. Eliminate choice E.

The correct answer is choice C.



Hi!
If the answer is (x+16)(x-11)= x^2 + 5x -176. Here, m= 176. Hence there can be multiple values for m. so answer should be E?
can anyone tell me what is wrong?
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Re: In the equation x^2+kxm=0 where k and m are both integers, what is m? [#permalink] New post   17 Jul 2022, 08:31
Urvasi wrote:
Abhishekgmat87 wrote:
Official Solution -

Evaluate Statement (1). Since there are variables, Plug In. If k = 5 and x = 2, then the equation in the question stem is \(2^2\) + (5)(2) – m = 0 which simplifies to 4 + 10 – m = 0 or 14 – m = 0 and m = 14. However, if k = 5 and x = 6, then the equation is 62 + (5)(6) – m = 0 which simplifies to 36 + 30 – m = 0 or 66 – m = 0 and m = 66. When different numbers that satisfy a statement yield different answers to the question, the statement is insufficient. So, write down BCE.

Now, evaluate Statement (2). It states that if both roots are negative, then k = 11. If the two roots are negative, then the two factors are positive. Since k is the sum of the last terms in the two factors and m is the product, the factors could be (x + 2)(x + 9). In that case, m = 2 × 9 =18. However, if the two factors are (x + 3)(x + 8), then m = 3 × 8 = 24. When different numbers that satisfy a statement yield different answers to the question, the statement is insufficient. Eliminate choice B.

Now, evaluate both statements together. Statement (2) states that if both roots are negative, then k = 11 and Statement (1) states that k = 5. So, in order to satisfy Statement (1), one of the roots is positive and one is negative because if both factors were positive, then the last terms would have to be negative. This means that the sum of the absolute value of the two last terms is 11 and the difference is 5. The only values that satisfy both statements are the factors (x + 8)(x – 3). Therefore, the original equation is \(x^2\) + 5x – 24 = 0 and m = 24. Since both statements together provide one specific answer to the question, which is 24, the statements together are sufficient. Eliminate choice E.

The correct answer is choice C.



Hi!
If the answer is (x+16)(x-11)= x^2 + 5x -176. Here, m= 176. Hence there can be multiple values for m. so answer should be E?
can anyone tell me what is wrong?



Sum of roots = -k. In your case sum of roots is 5 (-k)
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Re: In the equation x^2+kxm=0 where k and m are both integers, what is m? [#permalink] New post   28 Jul 2022, 08:30
1. If k=11, m = 24, we have the equation x^2+11x-24 = 0 that doesn't satisfy both roots are negative.
2. " in order to satisfy Statement (1), one of the roots is positive and one is negative...": the given equation can also have no root or one root with k=5.

Abhishekgmat87 wrote:
Official Solution -

Evaluate Statement (1). Since there are variables, Plug In. If k = 5 and x = 2, then the equation in the question stem is \(2^2\) + (5)(2) – m = 0 which simplifies to 4 + 10 – m = 0 or 14 – m = 0 and m = 14. However, if k = 5 and x = 6, then the equation is 62 + (5)(6) – m = 0 which simplifies to 36 + 30 – m = 0 or 66 – m = 0 and m = 66. When different numbers that satisfy a statement yield different answers to the question, the statement is insufficient. So, write down BCE.

Now, evaluate Statement (2). It states that if both roots are negative, then k = 11. If the two roots are negative, then the two factors are positive. Since k is the sum of the last terms in the two factors and m is the product, the factors could be (x + 2)(x + 9). In that case, m = 2 × 9 =18. However, if the two factors are (x + 3)(x + 8), then m = 3 × 8 = 24. When different numbers that satisfy a statement yield different answers to the question, the statement is insufficient. Eliminate choice B.

Now, evaluate both statements together. Statement (2) states that if both roots are negative, then k = 11 and Statement (1) states that k = 5. So, in order to satisfy Statement (1), one of the roots is positive and one is negative because if both factors were positive, then the last terms would have to be negative. This means that the sum of the absolute value of the two last terms is 11 and the difference is 5. The only values that satisfy both statements are the factors (x + 8)(x – 3). Therefore, the original equation is \(x^2\) + 5x – 24 = 0 and m = 24. Since both statements together provide one specific answer to the question, which is 24, the statements together are sufficient. Eliminate choice E.

The correct answer is choice C.
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In the equation x^2+kxm=0 where k and m are both integers, what is m? [#permalink] New post   28 Jul 2022, 16:43
Grant11 wrote:
1. If k=11, m = 24, we have the equation x^2+11x-24 = 0 that doesn't satisfy both roots are negative.
2. " in order to satisfy Statement (1), one of the roots is positive and one is negative...": the given equation can also have no root or one root with k=5.


Both roots in the equation x^2+11x-24 = 0 are negative. x = -3, -8
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Re: In the equation x^2+kxm=0 where k and m are both integers, what is m? [#permalink] New post   13 Aug 2022, 00:04
Abhishekgmat87 wrote:
Official Solution -

Evaluate Statement (1). Since there are variables, Plug In. If k = 5 and x = 2, then the equation in the question stem is \(2^2\) + (5)(2) – m = 0 which simplifies to 4 + 10 – m = 0 or 14 – m = 0 and m = 14. However, if k = 5 and x = 6, then the equation is 62 + (5)(6) – m = 0 which simplifies to 36 + 30 – m = 0 or 66 – m = 0 and m = 66. When different numbers that satisfy a statement yield different answers to the question, the statement is insufficient. So, write down BCE.

Now, evaluate Statement (2). It states that if both roots are negative, then k = 11. If the two roots are negative, then the two factors are positive. Since k is the sum of the last terms in the two factors and m is the product, the factors could be (x + 2)(x + 9). In that case, m = 2 × 9 =18. However, if the two factors are (x + 3)(x + 8), then m = 3 × 8 = 24. When different numbers that satisfy a statement yield different answers to the question, the statement is insufficient. Eliminate choice B.

Now, evaluate both statements together. Statement (2) states that if both roots are negative, then k = 11 and Statement (1) states that k = 5. So, in order to satisfy Statement (1), one of the roots is positive and one is negative because if both factors were positive, then the last terms would have to be negative. This means that the sum of the absolute value of the two last terms is 11 and the difference is 5. The only values that satisfy both statements are the factors (x + 8)(x – 3). Therefore, the original equation is \(x^2\) + 5x – 24 = 0 and m = 24. Since both statements together provide one specific answer to the question, which is 24, the statements together are sufficient. Eliminate choice E.

The correct answer is choice C.



can anyone tell me that why “ This means that the sum of the absolute value of the two last terms is 11 and the difference is 5.”

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Re: In the equation x^2+k*x-m=0 where k and m are both integers, what is [#permalink] New post   28 Aug 2022, 07:30
Official Answer Explanation:
Evaluate Statement (1). Since there are variables, Plug-In. If k = 5 and x = 2, then the equation in the question stem is 2^2 + (5)(2) – m = 0 which simplifies to 4 + 10 – m = 0 or 14 – m = 0 and m = 14. However, if k = 5 and x = 6, then the equation is 6^2 + (5)(6) – m = 0 which simplifies to 36 + 30 – m = 0 or 66 – m = 0 and m = 66. When different numbers that satisfy a statement yield different answers to the question, the statement is insufficient. So, write down BCE.

Now, evaluate Statement (2). It states that if both roots are negative, then k = 11. If the two roots are negative, then the two factors are positive. Since k is the sum of the last terms in the two factors and m is the product, the factors could be (x + 2)(x + 9). In that case, m = 2 × 9 =18. However, if the two factors are (x + 3)(x + 8), then m = 3 × 8 = 24. When different numbers that satisfy a statement yield different answers to the question, the statement is insufficient. Eliminate choice B.

Now, evaluate both statements together. Statement (2) states that if both roots are negative, then k = 11 and Statement (1) states that k = 5. So, in order to satisfy Statement (1), one of the roots is positive and one is negative because if both factors were positive, then the last terms would have to be negative. This means that the sum of the absolute value of the two last terms is 11 and the difference is 5. The only values that satisfy both statements are the factors (x + 8)(x – 3). Therefore, the original equation is x^2 + 5x – 24 = 0 and m = 24. Since both statements together provide one specific answer to the question, which is 24, the statements together are sufficient. Eliminate choice E. The correct answer is choice C.

Can someone simplify this or provide a better explanation as I am unable to understand this?
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Re: In the equation x^2+k*x-m=0 where k and m are both integers, what is [#permalink]
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