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If the quadratic equations 3x2 + ax + 1 = 0 and 2x2 + bx +1 = 0 have a common root, then the value of the expression 5ab – 2a2 – 3b2 is
- a)0
- b)1
- c)–1
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
If the quadratic equations 3x2+ ax + 1 = 0 and 2x2+ bx +1 = 0 have a c...
Let the common root be k. Then, we have:
3k^2 + ak + 1 = 0 ...(1)
2k^2 + bk + 1 = 0 ...(2)
Multiplying equation (1) by 2 and equation (2) by 3, we get:
6k^2 + 2ak + 2 = 0
6k^2 + 3bk + 3 = 0
Subtracting the second equation from the first, we get:
2ak - 3bk - 1 = 0
2ak = 3bk + 1
5ab = 5(a/2)(b/3) = (a/2)(3b/5)
Substituting 2ak = 3bk + 1, we get:
5ab = (a/2)(2ak/3) = (a/3)(k)
Substituting the value of k from equation (1), we get:
5ab = (a/3)(-ak - 1) = -a^2/3 - a/3
Multiplying by -3, we get:
-15ab = a^2 + a
Now, substituting the value of k from equation (2), we get:
5ab = (b/3)(-bk - 1) = -b^2/3 - b/3
Multiplying by -3, we get:
-15ab = b^2 + b
Adding the two equations, we get:
-30ab = a^2 + 2ab + b^2 + a + b
Simplifying, we get:
-30ab = (a + b)^2 + (a + b)
Substituting x = a + b, we get:
-30ab = x^2 + x
We are asked to find the value of 5ab, which is:
5ab = -x(x + 1)/30
Substituting x = -1, we get:
5ab = 1/30
Therefore, the value of the expression 5ab is 1/30.
3k^2 + ak + 1 = 0 ...(1)
2k^2 + bk + 1 = 0 ...(2)
Multiplying equation (1) by 2 and equation (2) by 3, we get:
6k^2 + 2ak + 2 = 0
6k^2 + 3bk + 3 = 0
Subtracting the second equation from the first, we get:
2ak - 3bk - 1 = 0
2ak = 3bk + 1
5ab = 5(a/2)(b/3) = (a/2)(3b/5)
Substituting 2ak = 3bk + 1, we get:
5ab = (a/2)(2ak/3) = (a/3)(k)
Substituting the value of k from equation (1), we get:
5ab = (a/3)(-ak - 1) = -a^2/3 - a/3
Multiplying by -3, we get:
-15ab = a^2 + a
Now, substituting the value of k from equation (2), we get:
5ab = (b/3)(-bk - 1) = -b^2/3 - b/3
Multiplying by -3, we get:
-15ab = b^2 + b
Adding the two equations, we get:
-30ab = a^2 + 2ab + b^2 + a + b
Simplifying, we get:
-30ab = (a + b)^2 + (a + b)
Substituting x = a + b, we get:
-30ab = x^2 + x
We are asked to find the value of 5ab, which is:
5ab = -x(x + 1)/30
Substituting x = -1, we get:
5ab = 1/30
Therefore, the value of the expression 5ab is 1/30.
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If the quadratic equations 3x2+ ax + 1 = 0 and 2x2+ bx +1 = 0 have a c...
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If the quadratic equations 3x2+ ax + 1 = 0 and 2x2+ bx +1 = 0 have a common root, then the value of the expression 5ab –2a2–3b2isa)0b)1c)–1d)None of theseCorrect answer is option 'B'. Can you explain this answer?
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