If the roots of the equation \(x^2 + mx + n = 0\) are increased by the same quantity \(k\), then they become the roots of the equation \(x^2 + nx + m = 0\). What is the value of \((m+n)\)?
-4
Let the roots of \(x^2+mx+n=0\) be \(p\) and \(q\), so \(p+q=-m\) and \(pq=n\).
The new roots \(p+k\) and \(q+k\) satisfy \(x^2+nx+m=0\), so \((p+k)+(q+k)=-n\) and \((p+k)(q+k)=m\).
From the sum: \(-m+2k=-n \Rightarrow 2k=m-n\).
From the product: \(n-km+k^2=m\). Substituting \(k=\frac{m-n}{2}\) and simplifying gives \((n-m)(n+m+4)=0\).
Since \(k\) is a genuine (non-zero) shift, \(n \neq m\), so \(m+n=-4\).
Similar Questions
Suppose \(p\) and \(q\) are positive integers where \(q\) is not a perfect square. If \(\dfrac{1}{p+\sqrt{q}}\) is a root of a quadratic equation having integer coefficients, then which of the following statements is/are correct?
I. The sum of the roots is \(2p\) times the product of the roots.
II. The sum of the reciprocals of the roots is equal to \(2p\).
Select the answer using the code given below:
If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^2 - 2(n+1)x + (n^2+n+1) = 0\), then what is \((2\alpha^3 + 2\beta^3)\) equal to?
Important Questions from Sum and Product of Roots
If the roots of the equation 4x 2- (5k + 1)x + 5k = 0 differ by unity, then which one of the following is a possible value of k?
For how many quadratic equations, the sum of roots is equal to the product of roots?
If α and β are the roots of the equation 4x 2+ 2x - 1 = 0, then which one of the following is correct?
What is sin (α + β) sec α sec β equal to?
What is tan (α + β) equal to?
