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If the roots of the equation ax2 + bx + c = 0 are real and distinct, then
- a)both roots are greater than -b/2a
- b)both roots are less than -b/2a
- c)one of the roots exceeds -b/2a
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
If the roots of the equation ax2+ bx + c = 0 are real and distinct, th...
Solution:
Real and distinct roots of the given quadratic equation ax2 + bx + c = 0 are given by:
x = [-b ± √(b2 - 4ac)]/2a
Let us assume that the roots of the given equation are α and β, such that α < />
Therefore, α + β = -b/a < />
Also, α × β = c/a > 0
We know that the sum of roots of a quadratic equation is -b/a, and the product of roots is c/a.
Now, we need to find out whether one of the roots exceeds -b/2a.
Let us assume that α > -b/2a.
Then, β > -b/2a (because α < />
α + β > -b/a
2α > -b/a
2αa > -b
-4ac > -b2
b2 - 4ac < />
This contradicts the fact that the given quadratic equation has real and distinct roots.
Therefore, α cannot be greater than -b/2a.
Similarly, we can prove that β cannot be less than -b/2a.
Hence, the correct option is C, i.e., one of the roots exceeds -b/2a.
Real and distinct roots of the given quadratic equation ax2 + bx + c = 0 are given by:
x = [-b ± √(b2 - 4ac)]/2a
Let us assume that the roots of the given equation are α and β, such that α < />
Therefore, α + β = -b/a < />
Also, α × β = c/a > 0
We know that the sum of roots of a quadratic equation is -b/a, and the product of roots is c/a.
Now, we need to find out whether one of the roots exceeds -b/2a.
Let us assume that α > -b/2a.
Then, β > -b/2a (because α < />
α + β > -b/a
2α > -b/a
2αa > -b
-4ac > -b2
b2 - 4ac < />
This contradicts the fact that the given quadratic equation has real and distinct roots.
Therefore, α cannot be greater than -b/2a.
Similarly, we can prove that β cannot be less than -b/2a.
Hence, the correct option is C, i.e., one of the roots exceeds -b/2a.
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If the roots of the equation ax2+ bx + c = 0 are real and distinct, thena)both roots are greater than -b/2ab)both roots are less than-b/2ac)one of the roots exceeds-b/2ad)none of theseCorrect answer is option 'C'. Can you explain this answer?
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