Let $p, q$ be the roots of the equation $x^2 + mx - n = 0$ and $m, n$ be the roots of the equation $x^2 + px - q = 0$ ($m, n, p, q$ are non-zero numbers). Which of the following statements is/are correct?
I. $m(m + n) = -1$
II. $p + q = 1$
Select the answer using the code given below:
The problem gives us two quadratic equations with their roots being mutually defined:
- The roots of the equation x^2 + mx - n = 0 are p and q.
- The roots of the equation x^2 + px - q = 0 are m and n.
Let's analyze the statements:
Statement I: m(m+n) = -1
Using Vieta's formulas for the first equation x^2 + mx - n = 0:
- p + q = -m
- pq = -n
From the second equation x^2 + px - q = 0 using Vieta's formulas again:
- m + n = -p
- mn = -q
We have the following expressions:
- p + q = -m
- pq = -n
- m + n = -p
- mn = -q
Rewriting the expression for m(m+n):
- m(m+n) = m(-p) = -mp
We also know from the roots of the equation that:
- m = \frac{q}{n}
- n = pq
- Combining with mn = -q, we have m = \frac{q}{n} and n = -\frac{q}{m}
- Thus, m(m+n) = m \left( - \frac{q}{m} \right) = -q and from the initial mn = -q, it implies that m(m+n) = mn.
- If m(m+n) = -1, we then conclude that mn = -1.
This confirms that Statement I: m(m+n) = -1 is correct.
Statement II: p + q = 1
We know from Vieta's formulas that p + q = -m. Without additional constraints relating m to 1, we cannot conclusively say that p + q = 1. Therefore, this statement is not necessarily true given the available information.
Hence, only Statement I is correct.
Conclusion:
Therefore, the correct answer is I only.
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Select the answer using the code given below : - What is the value of \(k\) for which \((k^2-5k+4)x^2+(k^2-3k-4)x+(k^2-4k)=0\) is an identity ?
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Find the minimum value of 2x² – 5x – 3 and also find x.
A quadratic equation $x^2 + 3x + k = 0$ has a discriminant equal to 9. What is the value of k?
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If both roots of the quadratic equation, $x^2 + 3x + 2 = 0$, are also roots of another quadratic equation, $ax^2 + bx + c = 0$, then which of the following must be true?
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