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If P(x) = ax2 + bx + c and Q(x) = -ax2 + dx + c, where ac ≠ 0, then P(x)Q(x) = 0 has
- a)no real root
- b)exactly two real roots
- c)at least two distinct real roots
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If P(x) = ax2 + bx + c and Q(x) = -ax2 + dx + c, where ac ≠ 0, then...
Let D1 = b2 – 4ac and D2 = d2 + 4ac
As ac ≠ 0, either ac < 0 or ac > 0
If ac < 0, then D1 > 0
In this case P(x) = 0 has distinct two real roots
If ac > 0, the D2 > 0
In this case Q(x) = 0 has two distinct real roots.
Thus, in either case P(x)Q(x) = 0 has at least two distinct real roots.
As ac ≠ 0, either ac < 0 or ac > 0
If ac < 0, then D1 > 0
In this case P(x) = 0 has distinct two real roots
If ac > 0, the D2 > 0
In this case Q(x) = 0 has two distinct real roots.
Thus, in either case P(x)Q(x) = 0 has at least two distinct real roots.
Most Upvoted Answer
If P(x) = ax2 + bx + c and Q(x) = -ax2 + dx + c, where ac ≠ 0, then...
And ac ≠ 0, then the sum of P(x) and Q(x) is:
P(x) + Q(x) = (ax^2 - bx + c) + (-ax^2 + dx + c)
= ax^2 - bx + c - ax^2 + dx + c
= (ax^2 - ax^2) + (-bx + dx) + (c + c)
= 0 + (d - b)x + 2c
Therefore, the sum of P(x) and Q(x) is (d - b)x + 2c.
P(x) + Q(x) = (ax^2 - bx + c) + (-ax^2 + dx + c)
= ax^2 - bx + c - ax^2 + dx + c
= (ax^2 - ax^2) + (-bx + dx) + (c + c)
= 0 + (d - b)x + 2c
Therefore, the sum of P(x) and Q(x) is (d - b)x + 2c.
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If P(x) = ax2 + bx + c and Q(x) = -ax2 + dx + c, where ac ≠ 0, then P(x)Q(x) = 0 hasa)no real rootb)exactly two real rootsc)at least two distinct real rootsd)none of theseCorrect answer is option 'C'. Can you explain this answer?
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