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$f(x)=x^{2}+bx+c$ , where $b,c \:\varepsilon \:R$ ,if $f(x)$ is a factor of both $x^{4}+6x^{2}+25\:\:and\:\:3x^{4}+4x^{2}+28x+5$ then the least value of $f(x)$ is
Option: 1 2

Option: 2 3

Option: 3 2.5

Option: 4 4

Answers (1)

best_answer

Quadratic Expression -

$f\left ( x \right )= ax^{2}+bx+c$

- wherein

$a\neq 0$

$a,b,c\in R,\: \:$

Condition for both roots common -

$\frac{a}{{a}'}=\frac{b}{{b}'}=\frac{c}{{c}'}$

- wherein

$ax^{2}+bx+c=0$ &

$a'x^{2}+b'x+c'=0$

are the 2 equations

$(x^{2}+bx+c).P(x)=3x^{4}+18x^{2}+75....................(i)$

$(x^{2}+bx+c).Q(x)=3x^{4}+4x^{2}+28x+5....................(ii)$

equation (i) – (ii)

$(x^{2}+bx+c).(P(x)-Q(x))=14x^{2}+28x+70$

$=14(x^{2}-2x+5)$

$x^{2}+bx+c=x^{2}-2x+5$

hence $f(x)=x^{2}-2x+5$

$=(x-1)^{2}+4$

$min\:\:\left ( f(x) \right )=4$

Posted by

Ritika Jonwal

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