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Number of integer values of $'a'$ for which $x^{2}+2\left ( a-1 \right )x+a\geqslant -5$ $\forall$ $x\: \epsilon \: R$ is

Option 1)
$6$

Option 2)
$5$

Option 3)
$4$

Option 4)
$3$

Answers (1)

best_answer

$x^{2}+2\left ( a-1 \right )x+\left ( a+5 \right )\geqslant 0\: \forall \: n\, \epsilon \, R$

So $D\leqslant 0$ is only required condition here as coefficient of $x^{2}$ is already positive.

$4\left ( a-1 \right )^{2}-4\left ( a+5 \right )\leqslant 0\: \Rightarrow \: \left ( a-4 \right )\left ( a+1 \right )\leqslant 0$

$\Rightarrow \: a\, \epsilon \, \left [ -1,4 \right ]$

$\therefore$ Option (A)

Quadratic Expression ax^2 + bx + c is non negative -

$ax^{2}+bx+c\geqslant 0$ for all $x \epsilon R$ When $a> 0$ & $b^{2}-4ac\leq 0$ $\left ( \; a,b,c\; \epsilon\; R \right )$

-

Option 1)

$6$

This is correct

Option 2)

$5$

This is incorrect

Option 3)

$4$

This is incorrect

Option 4)

$3$

This is incorrect

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