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09 Apr 2022, 08:25
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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29% (02:37) correct
71%
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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.
(1) k = 5
(2) If both roots of the equation were negative, the value of k would be 11.
Updated on: 08 Jul 2022, 09:07
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.
Last edited by Abhishekgmat87 on 08 Jul 2022, 09:07, edited 1 time in total.
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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.
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.
08 Jul 2022, 03:16
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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
Bunuel nick1816
