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Standard XII

Mathematics

Definition of Circle

Express x31...

Question

Express $x_{1}^{3} + x_{2}^{3}$ in terms of the coefficients of the equation $x^{2} + p x + q = 0,$ where $x_{1}$ and $x_{2}$ are the roots of the equation.

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Solution

$x^{2} + p x + q = 0$
Sum $= x_{1} + x_{2} = - p$
Product $= x_{1} x_{2} = q$
$x_{1}^{3} + x_{2}^{3} = (x_{1} + x_{2})^{3} - 3 x_{1} x_{2} (x_{1} + x_{2})$
$= (- p)^{3} - 3 q . (- p)$
$= 3 p q - p^{3}$

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Express x31+x32 in terms of the coefficients of the equation x2+px+ q=0, where x1 and x2 are the roots of the equation.

x2+px+q=0Sum =x1+x2=−pProduct =x1x2=qx31+x32=(x1+x2)3−3x1x2(x1+x2)=(−p)3−3q.(−p)=3pq−p3

Express $x_{1}^{3} + x_{2}^{3}$ in terms of the coefficients of the equation $x^{2} + p x + q = 0,$ where $x_{1}$ and $x_{2}$ are the roots of the equation.

$x^{2} + p x + q = 0$
Sum $= x_{1} + x_{2} = - p$
Product $= x_{1} x_{2} = q$
$x_{1}^{3} + x_{2}^{3} = (x_{1} + x_{2})^{3} - 3 x_{1} x_{2} (x_{1} + x_{2})$
$= (- p)^{3} - 3 q . (- p)$
$= 3 p q - p^{3}$