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Standard XII

Mathematics

Relations between Roots and Coefficients : Higher Order Equations

Equations x...

Question

Equations $x^{3} + 5 x^{2} + p x + q = 0$ and $x^{3} + 7 x^{2} + p x + r = 0$ have two roots in common. If the third root of each equation is $x_{1}$ and $x_{2}$ , respectively, then find $- (x_{1} + x_{2})$ ?

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Solution

Let roots of $x^{3} + 5 x^{2} + p x + q = 0$ be $α, β, x_{1}$ . Then
$α + β + x_{1} = - 5$
$α β + β x_{1} + α x_{1} = p$ ....(1)
Roots of $x^{3} + 7 x^{2} + p x + r = 0$ will be $α, β, x_{2}$ . Then
$α + β + x_{2} = - 7$
$α β + β x_{2} + α x_{2} = p$ ....(2)
Subtracting (2) from (1), we get
$(x_{1} - x_{2}) (α + β) = 0 (x_{1} \neq x_{2})$
$∴ α + β = 0$
$\Rightarrow x_{1} = - 5$ and $x_{2} = - 7$

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Equations x3+5x2+px+q=0 and x3+7x2+px+r=0 have two roots in common. If the third root of each equation is x1 and x2, respectively, then find −(x1+x2)?

Let roots of x3+5x2+px+q=0 be α,β,x1. Thenα+β+x1=−5αβ+βx1+αx1=p ....(1)Roots of x3+7x2+px+r=0 will be α,β,x2. Thenα+β+x2=−7αβ+βx2+αx2=p ....(2)Subtracting (2) from (1), we get(x1−x2)(α+β)=0(x1≠x2)∴α+β=0⇒x1=−5 and x2=−7

Equations $x^{3} + 5 x^{2} + p x + q = 0$ and $x^{3} + 7 x^{2} + p x + r = 0$ have two roots in common. If the third root of each equation is $x_{1}$ and $x_{2}$ , respectively, then find $- (x_{1} + x_{2})$ ?