The quadratic expression 2 x -12- px - q -0 for any real x if A- p2-16 p-8 q-0B- p 2-8 p -16 q -0C- p 2-16 p -8 q -0D- p2-8 p-16 q-0

The correct option is B p^2 8p 16q lt; 0(2x+1)^2 px + q = 0 ⇒ 4x^2+1+4x px +q = 0 ⇒ 4x^2+ (4 p)x+ (1+q) = 0 If nbsp;D lt; 0, then the quadratic expres ...

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Standard XII

Mathematics

Local Maxima

The quadratic...

Question

The quadratic expression $(2 x + 1)^{2} - p x + q \neq 0$ for any real $x$ if

p^{2} - 16 p - 8 q < 0

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p^{2} - 8 p + 16 q < 0

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p^{2} - 8 p - 16 q < 0

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p^{2} - 16 p + 8 q < 0

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Solution

The correct option is C $p^{2} - 8 p - 16 q < 0$
$(2 x + 1)^{2} - p x + q = 0$
$\Rightarrow 4 x^{2} + 1 + 4 x - p x + q = 0 \Rightarrow 4 x^{2} + (4 - p) x + (1 + q) = 0$
If $D < 0$ , then the quadratic expression $(2 x + 1)^{2} - p x + q \neq 0$ holds true.

$∴ (4 - p)^{2} - 16 (1 + q) < 0$
$\Rightarrow p^{2} - 8 p - 16 q < 0$

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The quadratic expression (2x+1)2−px+q≠0 for any real x if

The correct option is C p2−8p−16q<0(2x+1)2−px+q=0 ⇒4x2+1+4x−px+q=0⇒4x2+(4−p)x+(1+q)=0 If D<0, then the quadratic expression (2x+1)2−px+q≠0 holds true. ∴(4−p)2−16(1+q)<0 ⇒p2−8p−16q<0

The quadratic expression $(2 x + 1)^{2} - p x + q \neq 0$ for any real $x$ if