If the roots of x 2- mx - n -0 are twice the roots of x 2- px - m -0- p -0- then the value of n - p is

Let the roots of x^2+px+m=0 be α,β α+β= p and αβ=m ⋯(1) Roots of nbsp;x^2+mx+n=0 are nbsp;2α,2β nbsp; ⇒ 2α+2β= m and 2α·2β=n ⇒ 2(α+β)= m and 4αβ=n ...

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Standard XII

Mathematics

Roots of a Quadratic Equation

If the roots ...

Question

If the roots of $x^{2} + m x + n = 0$ are twice the roots of $x^{2} + p x + m = 0, p \neq 0$ , then the value of $\frac{n}{p}$ is

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Solution

Let the roots of $x^{2} + p x + m = 0$ be $α, β$
$α + β = - p$ and $α β = m \dots (1)$

Roots of $x^{2} + m x + n = 0$ are $2 α, 2 β$
$\Rightarrow 2 α + 2 β = - m$ and $2 α \cdot 2 β = n$
$\Rightarrow 2 (α + β) = - m$ and $4 α β = n$
$\Rightarrow 2 p = m$ and $4 m = n [Using (1)]$
$\Rightarrow 8 p = n$
$\Rightarrow \frac{n}{p} = 8$

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If the roots of x2+mx+n=0 are twice the roots of x2+px+m=0,p≠0, then the value of np is

Let the roots of x2+px+m=0 be α,β α+β=−p and αβ=m ⋯(1) Roots of x2+mx+n=0 are 2α,2β ⇒2α+2β=−m and 2α⋅2β=n ⇒2(α+β)=−m and 4αβ=n ⇒2p=m and 4m=n [Using (1)] ⇒8p=n ⇒np=8

If the roots of $x^{2} + m x + n = 0$ are twice the roots of $x^{2} + p x + m = 0, p \neq 0$ , then the value of $\frac{n}{p}$ is