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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)2
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 ...
6x2 + 2ax + 2 = 0 and 6x2 + 3bx + 3 = 0 subtracting x (2a – 3b) – 1 = 0
(put in any equation)
2 + b (2a – 3b) + (2a – 3b)2 = 0
4a2 + 5b2 – 12ab + 2ab – 3b2 + 2 = 0
–10ab + 6b2 + 4a2 + 1 = 0
⇒ 5ab –3b2 – 2a2 = 1 ⇒ B
(put in any equation)
2 + b (2a – 3b) + (2a – 3b)2 = 0
4a2 + 5b2 – 12ab + 2ab – 3b2 + 2 = 0
–10ab + 6b2 + 4a2 + 1 = 0
⇒ 5ab –3b2 – 2a2 = 1 ⇒ B
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Community Answer
If the quadratic equations 3x2 + ax + 1 = 0 and 2x2 + bx + 1 = 0 have ...
Common Root of Quadratic Equations
When two quadratic equations have a common root, it means that this common root satisfies both equations simultaneously. In this case, the common root satisfies both 3x^2 + ax + 1 = 0 and 2x^2 + bx + 1 = 0.
Using Vieta's Formulas
Vieta's formulas state that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a.
For the first equation, the sum of the roots is -a/3 and the product of the roots is 1/3.
For the second equation, the sum of the roots is -b/2 and the product of the roots is 1/2.
Finding the Common Root
Since the common root satisfies both equations, we can set them equal to each other and solve for x.
3x^2 + ax + 1 = 2x^2 + bx + 1
x^2 + (a-b)x = 0
x(x + a - b) = 0
This implies that either x = 0 or x = b - a.
Calculating the Expression
Now that we know the common root is either 0 or b - a, we can calculate the expression 5ab - 2a^2 - 3b^2.
If the common root is 0, then the expression simplifies to 0 - 2a^2 - 3b^2 = -2a^2 - 3b^2.
If the common root is b - a, then the expression simplifies to 5ab - 2a^2 - 3b^2.
Conclusion
Since the value of the expression can take on two different forms based on the common root, we can see that the final value will depend on the common root itself. However, it is clear that the value will be 0 if the common root is 0. Therefore, the correct answer is option B) 1.
When two quadratic equations have a common root, it means that this common root satisfies both equations simultaneously. In this case, the common root satisfies both 3x^2 + ax + 1 = 0 and 2x^2 + bx + 1 = 0.
Using Vieta's Formulas
Vieta's formulas state that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a.
For the first equation, the sum of the roots is -a/3 and the product of the roots is 1/3.
For the second equation, the sum of the roots is -b/2 and the product of the roots is 1/2.
Finding the Common Root
Since the common root satisfies both equations, we can set them equal to each other and solve for x.
3x^2 + ax + 1 = 2x^2 + bx + 1
x^2 + (a-b)x = 0
x(x + a - b) = 0
This implies that either x = 0 or x = b - a.
Calculating the Expression
Now that we know the common root is either 0 or b - a, we can calculate the expression 5ab - 2a^2 - 3b^2.
If the common root is 0, then the expression simplifies to 0 - 2a^2 - 3b^2 = -2a^2 - 3b^2.
If the common root is b - a, then the expression simplifies to 5ab - 2a^2 - 3b^2.
Conclusion
Since the value of the expression can take on two different forms based on the common root, we can see that the final value will depend on the common root itself. However, it is clear that the value will be 0 if the common root is 0. Therefore, the correct answer is option B) 1.
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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 isa)0b)1c)- 1d)2Correct answer is option 'B'. Can you explain this answer?
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