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If the zeroes of the quadratic polynomial ax2 + bx + c, c ≠ 0 are equal, then
- a)c and b have opposite signs
- b)c and a have opposite signs
- c)c and b have same signs
- d)c and a have same signs
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
If the zeroes of the quadratic polynomial ax2+ bx + c, c ≠ 0 are eq...
For equal roots, discriminant will be equal to zero.
b2 -4ac = 0
b2 = 4ac
ac = b2/4
ac > 0 (as square of any number cannot be negative)
Most Upvoted Answer
If the zeroes of the quadratic polynomial ax2+ bx + c, c ≠ 0 are eq...
Understanding Equal Roots in Quadratic Polynomials
In a quadratic polynomial of the form ax² + bx + c, the nature of the roots (zeroes) is determined by the discriminant (D), given by the formula D = b² - 4ac. For the roots to be equal, the discriminant must equal zero.
Condition for Equal Roots
- When D = 0, we have:
b² - 4ac = 0
- Rearranging this gives:
b² = 4ac
Analyzing the Signs of a, b, and c
Given that c ≠ 0 (as stated in the question), we can analyze the implications of the equation b² = 4ac:
1. b² is Always Non-Negative:
- Since b² is a square term, it is always greater than or equal to zero.
2. Implications for a and c:
- For the product 4ac to equal b² (which is non-negative), the product ac must also be non-negative.
- This means that both a and c must either be positive or both must be negative.
Conclusion
From the analysis above, we can conclude:
- Since c ≠ 0 and for b² to equal 4ac, a and c must have the same sign.
- Therefore, the correct answer is option 'D': c and a have the same signs.
This understanding reinforces the relationship between the coefficients of a quadratic polynomial and the nature of its roots.
In a quadratic polynomial of the form ax² + bx + c, the nature of the roots (zeroes) is determined by the discriminant (D), given by the formula D = b² - 4ac. For the roots to be equal, the discriminant must equal zero.
Condition for Equal Roots
- When D = 0, we have:
b² - 4ac = 0
- Rearranging this gives:
b² = 4ac
Analyzing the Signs of a, b, and c
Given that c ≠ 0 (as stated in the question), we can analyze the implications of the equation b² = 4ac:
1. b² is Always Non-Negative:
- Since b² is a square term, it is always greater than or equal to zero.
2. Implications for a and c:
- For the product 4ac to equal b² (which is non-negative), the product ac must also be non-negative.
- This means that both a and c must either be positive or both must be negative.
Conclusion
From the analysis above, we can conclude:
- Since c ≠ 0 and for b² to equal 4ac, a and c must have the same sign.
- Therefore, the correct answer is option 'D': c and a have the same signs.
This understanding reinforces the relationship between the coefficients of a quadratic polynomial and the nature of its roots.
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