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If a + b + c = 0 then the equation 3ax2 + 2bx + c = 0 has
- a)imaginary roots
- b)real and equal roots
- c)real and different roots
- d)rational roots
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If a + b + c = 0 then the equation 3ax2 + 2bx + c = 0 hasa)imaginary r...
Δ = 4(a2 + c2 - ac) = 2(a2 + c2 + (a - c2)) > 0
Most Upvoted Answer
If a + b + c = 0 then the equation 3ax2 + 2bx + c = 0 hasa)imaginary r...
Explanation:
Given: a + b + c = 0
Equation: 3ax^2 + 2bx + c = 0
Roots of a Quadratic Equation:
- For a quadratic equation of the form ax^2 + bx + c = 0, the roots can be real or imaginary depending on the discriminant (b^2 - 4ac).
- If the discriminant is greater than 0, the roots are real and different.
- If the discriminant is equal to 0, the roots are real and equal.
- If the discriminant is less than 0, the roots are imaginary.
Relationship between Coefficients and Roots:
- Given a + b + c = 0, if we substitute x = 1 in the equation, we get 3a + 2b + c = 0.
- Subtracting the two equations, we get 2a + b = 0.
- This implies b = -2a.
Substitute b = -2a into the quadratic equation:
- 3ax^2 - 4ax + c = 0
- Simplifying, we get 3ax^2 - 4ax - 2a = 0
- Factoring out an 'a', we get a(3x^2 - 4x - 2) = 0
Discriminant:
- For the equation 3x^2 - 4x - 2 = 0, the discriminant is 4^2 - 4(3)(-2) = 16 + 24 = 40, which is greater than 0.
Conclusion:
- Since the discriminant is greater than 0, the roots of the equation 3ax^2 + 2bx + c = 0 are real and different.
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If a + b + c = 0 then the equation 3ax2 + 2bx + c = 0 hasa)imaginary rootsb)real and equal rootsc)real and different rootsd)rational rootsCorrect answer is option 'C'. Can you explain this answer?
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