Consider x3-ax2-bx-c-0 where a-b-c are rationals and a- 1-It is given that x1- x2- x1x2 are roots of the above equation and if b-c-2a-1 then x1x2- -

x_1+x_2+x_1x_2=a x_1x_2+x_1^2x_2+x_2^2x_1=b (x_1x_2)^2=c Now, b+ca+1=2 ⇒x_1x_2+x_1^2x_2+x_2^2x_1+(x_1x_2)^2x_1+x_2+x_1x_2+1=2 ⇒x_1x ...

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Byju's Answer

Standard XII

Mathematics

Sign of Quadratic Expression

Consider x3...

Question

Consider $x^{3} - a x^{2} + b x - c = 0$ where $a, b, c$ are rationals and $a \neq 1$ .It is given that $x_{1}, x_{2}, x_{1} x_{2}$ are roots of the above equation and if $b + c = 2 (a + 1)$ then $x_{1} x_{2} =$ ______________.

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Solution

$x_{1} + x_{2} + x_{1} x_{2} = a$
$x_{1} x_{2} + x_{1}^{2} x_{2} + x_{2}^{2} x_{1} = b$
${(x_{1} x_{2})}^{2} = c$
Now, $\frac{b + c}{a + 1} = 2$
$\Rightarrow \frac{x_{1} x_{2} + x_{1}^{2} x_{2} + x_{2}^{2} x_{1} + {(x_{1} x_{2})}^{2}}{x_{1} + x_{2} + x_{1} x_{2} + 1} = 2$
$\Rightarrow \frac{x_{1} x_{2} (1 + x_{1} + x_{2} + x_{1} x_{2})}{1 + x_{1} + x_{2} + x_{1} x_{2}} = 2$
$\Rightarrow x_{1} x_{2} = 2$

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Consider x3−ax2+bx−c=0 where a,b,c are rationals and a≠1.It is given that x1,x2,x1x2 are roots of the above equation and if b+c=2(a+1) then x1x2= ______________.

x1+x2+x1x2=ax1x2+x21x2+x22x1=b(x1x2)2=cNow,b+ca+1=2⇒x1x2+x21x2+x22x1+(x1x2)2x1+x2+x1x2+1=2⇒x1x2(1+x1+x2+x1x2)1+x1+x2+x1x2=2⇒x1x2=2

Consider $x^{3} - a x^{2} + b x - c = 0$ where $a, b, c$ are rationals and $a \neq 1$ .It is given that $x_{1}, x_{2}, x_{1} x_{2}$ are roots of the above equation and if $b + c = 2 (a + 1)$ then $x_{1} x_{2} =$ ______________.