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A quadratic equation ax2 + bx + c = 0 has two integral roots x1 and x2. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following could be the value of the ordered set (a, b, c)?
I. (-1, 4, -3)
II. (1, 4, 3)
III. (3, -10√3, 9)
- a)I Only
- b)II Only
- c)III Only
- d)I and II Only
- e)I, II and III Only
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A quadratic equation ax2 + bx + c = 0 has two integral roots x1 and x2...
Given
To Find: Values of (a, b, c)
Approach
- For finding the values of (a, b, c), we would first need to find the value of x1 , x2
- We will use the relation
-
- to find out the values of x1 , x2
- Also, we will keep in mind the constraint that x1 , x2 are integers
- Now, we know that
- We will use the relation
- We will use the above relation to find out the possible values of (a, b, c)
As x1 , x2 are integers, the possible cases for (x1 , x2) is either (3,1) or (-3,-1)
- Using (1), (2) and (3), the values of (a, b, c) can be of the form ( a, 4a, 3a) or (a, -4a, 3a)
- Among the options,
- Option-I (-1, 4, -3) is of the form (a, -4a, 3a) and
- Option- II (1, 4, 3) is of the form (a, 4a, 3a)
Hence, options I and II can be the value of ordered set (a, b, c).
Answer: D
Most Upvoted Answer
A quadratic equation ax2 + bx + c = 0 has two integral roots x1 and x2...
To solve this question, we need to use the quadratic formula and the given information about the sum and squares of the roots.
The quadratic formula states that for a quadratic equation ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Let's first analyze the given information about the sum and squares of the roots. The square of the sum of the roots is given by:
(x1 + x2)^2 = (x1^2 + 2x1x2 + x2^2)
The sum of the squares of the roots is given by:
x1^2 + x2^2
According to the given information, the square of the sum of the roots is 6 greater than the sum of the squares of the roots, so we can write the equation:
(x1^2 + 2x1x2 + x2^2) = (x1^2 + x2^2) + 6
Simplifying this equation, we get:
2x1x2 = 6
Now let's analyze each option given and check if they satisfy this condition:
I. (-1, 4, -3)
Using the quadratic formula, we find the roots of the equation -x^2 + 4x - 3 = 0 as x1 = 1 and x2 = 3.
The sum of the roots is 1 + 3 = 4, and the sum of the squares of the roots is 1^2 + 3^2 = 10.
The square of the sum of the roots is (1 + 3)^2 = 16, which is not 6 greater than the sum of the squares of the roots.
Therefore, option I does not satisfy the given condition.
II. (1, 4, 3)
Using the quadratic formula, we find the roots of the equation x^2 + 4x + 3 = 0 as x1 = -1 and x2 = -3.
The sum of the roots is -1 + (-3) = -4, and the sum of the squares of the roots is (-1)^2 + (-3)^2 = 10.
The square of the sum of the roots is (-1 + (-3))^2 = 16, which is 6 greater than the sum of the squares of the roots.
Therefore, option II satisfies the given condition.
III. (3, -103, 9)
Using the quadratic formula, we find the roots of the equation 3x^2 - 103x + 9 = 0 as x1 = 1 and x2 = 9/3 = 3.
The sum of the roots is 1 + 3 = 4, and the sum of the squares of the roots is 1^2 + 3^2 = 10.
The square of the sum of the roots is (1 + 3)^2 = 16, which is 6 greater than the sum of the squares of the roots.
Therefore, option III satisfies the given condition.
Based on the above analysis, options II and III satisfy the given condition, so the correct answer is option D, "I and II Only."
The quadratic formula states that for a quadratic equation ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Let's first analyze the given information about the sum and squares of the roots. The square of the sum of the roots is given by:
(x1 + x2)^2 = (x1^2 + 2x1x2 + x2^2)
The sum of the squares of the roots is given by:
x1^2 + x2^2
According to the given information, the square of the sum of the roots is 6 greater than the sum of the squares of the roots, so we can write the equation:
(x1^2 + 2x1x2 + x2^2) = (x1^2 + x2^2) + 6
Simplifying this equation, we get:
2x1x2 = 6
Now let's analyze each option given and check if they satisfy this condition:
I. (-1, 4, -3)
Using the quadratic formula, we find the roots of the equation -x^2 + 4x - 3 = 0 as x1 = 1 and x2 = 3.
The sum of the roots is 1 + 3 = 4, and the sum of the squares of the roots is 1^2 + 3^2 = 10.
The square of the sum of the roots is (1 + 3)^2 = 16, which is not 6 greater than the sum of the squares of the roots.
Therefore, option I does not satisfy the given condition.
II. (1, 4, 3)
Using the quadratic formula, we find the roots of the equation x^2 + 4x + 3 = 0 as x1 = -1 and x2 = -3.
The sum of the roots is -1 + (-3) = -4, and the sum of the squares of the roots is (-1)^2 + (-3)^2 = 10.
The square of the sum of the roots is (-1 + (-3))^2 = 16, which is 6 greater than the sum of the squares of the roots.
Therefore, option II satisfies the given condition.
III. (3, -103, 9)
Using the quadratic formula, we find the roots of the equation 3x^2 - 103x + 9 = 0 as x1 = 1 and x2 = 9/3 = 3.
The sum of the roots is 1 + 3 = 4, and the sum of the squares of the roots is 1^2 + 3^2 = 10.
The square of the sum of the roots is (1 + 3)^2 = 16, which is 6 greater than the sum of the squares of the roots.
Therefore, option III satisfies the given condition.
Based on the above analysis, options II and III satisfy the given condition, so the correct answer is option D, "I and II Only."
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A quadratic equation ax2 + bx + c = 0 has two integral roots x1 and x2. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following couldbe the value of the ordered set (a, b, c)?I. (-1, 4, -3)II. (1, 4, 3)III. (3, -10√3, 9)a)I Onlyb)II Onlyc)III Onlyd)I and II Onlye)I, II andIII OnlyCorrect answer is option 'D'. Can you explain this answer?
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A quadratic equation ax2 + bx + c = 0 has two integral roots x1 and x2. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following couldbe the value of the ordered set (a, b, c)?I. (-1, 4, -3)II. (1, 4, 3)III. (3, -10√3, 9)a)I Onlyb)II Onlyc)III Onlyd)I and II Onlye)I, II andIII OnlyCorrect answer is option 'D'. 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 A quadratic equation ax2 + bx + c = 0 has two integral roots x1 and x2. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following couldbe the value of the ordered set (a, b, c)?I. (-1, 4, -3)II. (1, 4, 3)III. (3, -10√3, 9)a)I Onlyb)II Onlyc)III Onlyd)I and II Onlye)I, II andIII OnlyCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A quadratic equation ax2 + bx + c = 0 has two integral roots x1 and x2. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following couldbe the value of the ordered set (a, b, c)?I. (-1, 4, -3)II. (1, 4, 3)III. (3, -10√3, 9)a)I Onlyb)II Onlyc)III Onlyd)I and II Onlye)I, II andIII OnlyCorrect answer is option 'D'. Can you explain this answer?.
A quadratic equation ax2 + bx + c = 0 has two integral roots x1 and x2. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following couldbe the value of the ordered set (a, b, c)?I. (-1, 4, -3)II. (1, 4, 3)III. (3, -10√3, 9)a)I Onlyb)II Onlyc)III Onlyd)I and II Onlye)I, II andIII OnlyCorrect answer is option 'D'. 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 A quadratic equation ax2 + bx + c = 0 has two integral roots x1 and x2. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following couldbe the value of the ordered set (a, b, c)?I. (-1, 4, -3)II. (1, 4, 3)III. (3, -10√3, 9)a)I Onlyb)II Onlyc)III Onlyd)I and II Onlye)I, II andIII OnlyCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A quadratic equation ax2 + bx + c = 0 has two integral roots x1 and x2. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following couldbe the value of the ordered set (a, b, c)?I. (-1, 4, -3)II. (1, 4, 3)III. (3, -10√3, 9)a)I Onlyb)II Onlyc)III Onlyd)I and II Onlye)I, II andIII OnlyCorrect answer is option 'D'. Can you explain this answer?.
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