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In the equation ax2 + bx + c = 0, where a, b and c are constants and a ≠ 0, what is the value of b?
(1) 3 and 4 are roots of the equation.
(2) The product of the roots of the equation is 12.
- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- d)EACH statement ALONE is sufficient.
- e)Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'E'. Can you explain this answer?
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Verified Answer
In the equation ax2 + bx + c = 0, where a, b and c are constants and a...
Steps 1 & 2: Understand Question and Draw Inferences
Given equation is
ax2+bx+c=0
It is also given that a, b and c are constants and that a≠0.
We need to find the value of b.
Step 3: Analyze Statement 1
It is given that 3 and 4 are the roots of the equation.
We know that for a quadratic equation
ax2+bx+c=0
ax2+bx+c=0
Sum of roots = −b/a
Product of roots = c/a
Therefore, we have:
−ba = 3 + 4 = 7………..(I)
c/ a = 3 * 4 = 12 …………(II)
Note that these are two linear equations in three variables. So we cannot solve exclusively for b.
Statement 1 alone is not sufficient to arrive at a unique answer.
Step 4: Analyze Statement 2
It is given that the product of roots is 12.
Therefore, we have:
c/a = 12 …….(III)
Notice that (III) is a single linear equation in two variables. Moreover, it doesn’t provide any information about b.
So statement 2 alone is not sufficient to arrive at a unique answer.
Step 5: Analyze Both Statements Together (if needed)
Now let us look at both the statements together.
Statement 1 gives us (I) and (II)
Statement 2 gives us (III).
However, notice that (III) is essentially same as (II).
In other words, we are simply left with (I) and (II) (essentially the same situation as in statement 1).
Therefore statement 1 and statement 2 combined together are not sufficient to arrive at a unique answer.
Answer: Option (E)
Most Upvoted Answer
In the equation ax2 + bx + c = 0, where a, b and c are constants and a...
Understanding the Problem
To find the value of \( b \) in the equation \( ax^2 + bx + c = 0 \), we can use the relationships between the roots of the quadratic equation. The roots of the equation can be denoted as \( r_1 \) and \( r_2 \).
Key Relationships
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \)
- The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \)
Evaluating Statement (1)
- Statement (1): 3 and 4 are roots of the equation.
- From this, we can calculate:
- Sum of the roots: \( 3 + 4 = 7 \)
- Therefore, \( -\frac{b}{a} = 7 \) implies \( b = -7a \).
- However, without knowing the value of \( a \), we cannot determine a unique value for \( b \). Thus, statement (1) alone is insufficient.
Evaluating Statement (2)
- Statement (2): The product of the roots of the equation is 12.
- This gives us: \( r_1 \cdot r_2 = 12 \) implies \( \frac{c}{a} = 12 \).
- We cannot determine \( b \) as we still lack information about the sum of the roots. Thus, statement (2) alone is also insufficient.
Combining Statements (1) and (2)
- Combining both statements:
- From statement (1), we have \( b = -7a \).
- From statement (2), we know the product of the roots is 12, but we still lack the actual values of \( a \) and \( c \).
- Therefore, even together, the statements do not provide enough information to uniquely determine \( b \).
Conclusion
Thus, the correct answer is option E: Statements (1) and (2) together are NOT sufficient.
To find the value of \( b \) in the equation \( ax^2 + bx + c = 0 \), we can use the relationships between the roots of the quadratic equation. The roots of the equation can be denoted as \( r_1 \) and \( r_2 \).
Key Relationships
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \)
- The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \)
Evaluating Statement (1)
- Statement (1): 3 and 4 are roots of the equation.
- From this, we can calculate:
- Sum of the roots: \( 3 + 4 = 7 \)
- Therefore, \( -\frac{b}{a} = 7 \) implies \( b = -7a \).
- However, without knowing the value of \( a \), we cannot determine a unique value for \( b \). Thus, statement (1) alone is insufficient.
Evaluating Statement (2)
- Statement (2): The product of the roots of the equation is 12.
- This gives us: \( r_1 \cdot r_2 = 12 \) implies \( \frac{c}{a} = 12 \).
- We cannot determine \( b \) as we still lack information about the sum of the roots. Thus, statement (2) alone is also insufficient.
Combining Statements (1) and (2)
- Combining both statements:
- From statement (1), we have \( b = -7a \).
- From statement (2), we know the product of the roots is 12, but we still lack the actual values of \( a \) and \( c \).
- Therefore, even together, the statements do not provide enough information to uniquely determine \( b \).
Conclusion
Thus, the correct answer is option E: Statements (1) and (2) together are NOT sufficient.
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In the equation ax2 + bx + c = 0, where a, b and c are constants and a ≠ 0, what is the value of b?(1) 3 and 4 are roots of the equation.(2) The product of the roots of the equation is 12.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.d)EACH statement ALONE is sufficient.e)Statements (1) and (2) TOGETHER are NOT sufficient.Correct answer is option 'E'. Can you explain this answer?
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In the equation ax2 + bx + c = 0, where a, b and c are constants and a ≠ 0, what is the value of b?(1) 3 and 4 are roots of the equation.(2) The product of the roots of the equation is 12.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.d)EACH statement ALONE is sufficient.e)Statements (1) and (2) TOGETHER are NOT sufficient.Correct answer is option 'E'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about In the equation ax2 + bx + c = 0, where a, b and c are constants and a ≠ 0, what is the value of b?(1) 3 and 4 are roots of the equation.(2) The product of the roots of the equation is 12.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.d)EACH statement ALONE is sufficient.e)Statements (1) and (2) TOGETHER are NOT sufficient.Correct answer is option 'E'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the equation ax2 + bx + c = 0, where a, b and c are constants and a ≠ 0, what is the value of b?(1) 3 and 4 are roots of the equation.(2) The product of the roots of the equation is 12.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.d)EACH statement ALONE is sufficient.e)Statements (1) and (2) TOGETHER are NOT sufficient.Correct answer is option 'E'. Can you explain this answer?.
In the equation ax2 + bx + c = 0, where a, b and c are constants and a ≠ 0, what is the value of b?(1) 3 and 4 are roots of the equation.(2) The product of the roots of the equation is 12.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.d)EACH statement ALONE is sufficient.e)Statements (1) and (2) TOGETHER are NOT sufficient.Correct answer is option 'E'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about In the equation ax2 + bx + c = 0, where a, b and c are constants and a ≠ 0, what is the value of b?(1) 3 and 4 are roots of the equation.(2) The product of the roots of the equation is 12.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.d)EACH statement ALONE is sufficient.e)Statements (1) and (2) TOGETHER are NOT sufficient.Correct answer is option 'E'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the equation ax2 + bx + c = 0, where a, b and c are constants and a ≠ 0, what is the value of b?(1) 3 and 4 are roots of the equation.(2) The product of the roots of the equation is 12.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.d)EACH statement ALONE is sufficient.e)Statements (1) and (2) TOGETHER are NOT sufficient.Correct answer is option 'E'. Can you explain this answer?.
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