Question

X^2 + x/√2 + 1=0 Class 11 complex num

Answer 1

$\large\underline{\sf{Solution-}}$

Given quadratic equation is

$\rm :\longmapsto\: {x}^{2} + \dfrac{x}{ \sqrt{2} } + 1 = 0$

can be rewritten as

$\rm :\longmapsto\: \sqrt{2} {x}^{2} + x + \sqrt{2} = 0$

So, on comparing with quadratic equation ax² + bx + c = 0, we get

$\red{\rm :\longmapsto\:a = \sqrt{2}}$

$\red{\rm :\longmapsto\:b = 1}$

$\red{\rm :\longmapsto\:c = \sqrt{2} }$

Let first evaluate the Discriminant, D of the quadratic equation which is given by

$\blue{\rm :\longmapsto\:D = {b}^{2} - 4ac \: }$

On substituting the values of a, b and c, we get

$\blue{\rm :\longmapsto\:D = {1}^{2} - 4( \sqrt{2}) \: ( \sqrt{2})\: }$

$\blue{\rm :\longmapsto\:D = 1 - 8\: }$

$\blue{\rm :\longmapsto\:D = - 7\: }$

Since, D < 0, it implies equation has imaginary or complex roots.

So, Roots of quadratic equation is given by

$\green{\rm :\longmapsto\:x = \dfrac{ - b \: \pm \: \sqrt{ {b}^{2} - 4ac}}{2a} \: }$

OR

$\green{\rm :\longmapsto\:x = \dfrac{ - b \: \pm \: \sqrt{D}}{2a} \: }$

So, on substituting the values of a, b and D, we get

$\green{\rm :\longmapsto\:x = \dfrac{ - 1 \: \pm \: \sqrt{ - 7}}{2 \sqrt{2} } \: }$

$\green{\rm :\longmapsto\:x = \dfrac{ - 1 \: \pm \: i \: \sqrt{7}}{2 \sqrt{2} } \: }$

$\green{\rm \implies\:\boxed{ \tt{ \: x = \dfrac{ - 1 + i \sqrt{7} }{2 \sqrt{2} } \: or \: \dfrac{ - 1 - i\sqrt{7} }{2 \sqrt{2} } }}}$

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Additional Information :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac

Answer 2

Question :-

$\rm{x {}^{2} + \frac{x}{ \sqrt{2} } + 1 = 0 }$ Class 11 complex number.

Answer :-

$\small{\boxed{\tt{x = \frac{ - 1 +i \sqrt{7 } }{2 \sqrt{2} } \: \: \: \: \: \: \: \: \: or \: \: \: \: \: \: \: \: \: x = \frac{ - 1 - i \sqrt{7} }{2 \sqrt{2} } }}}$

Step by step explanation :-

Given Equation :

$\rm:\longmapsto{x {}^{2} + \frac{x}{ \sqrt{2} } + 1 = 0 }$

Multiplying the equation by √2

$\rm:\longmapsto{ \sqrt{2}x {}^{2} + x + \sqrt{2} = 0 }$

Comparing the equation with ax² + bx + c,

• a = √2
• b = 1
• c = √2

The Quadratic Formula,

$\tt:\longmapsto{x = \frac{ - b \pm \sqrt{b {}^{2} - 4ac} }{2a} }$

Substituting a, b and c in the Quadratic formula,

$\rm:\longmapsto{x = \frac{ - 1 \pm \sqrt{(1) {}^{2} - 4( \sqrt{2} )( \sqrt{2})} }{2( \sqrt{2}) } } \\ \\\rm:\longmapsto{x = \frac{ - 1 \pm \sqrt{1 - 4(2)} }{ 2\sqrt{2} } } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm:\longmapsto{x = \frac{ - 1 \pm \sqrt{ - 7} }{2 \sqrt{2} } } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

We know that,

$\tt:\longmapsto{i = \sqrt{ - 1} }$

Hence,

$\: \small{\boxed{\tt{x = \frac{ - 1 +i \sqrt{7 } }{2 \sqrt{2} } \: \: \: \: \: \: \: \: \: or \: \: \: \: \: \: \: \: \: x = \frac{ - 1 - i \sqrt{7} }{2 \sqrt{2} } }}}$

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If a is one imaginery root of x^(2)-1=0 then the equation whone roos are a+a^(4) a^(2)+a' in A) x^(2)-x-1=0 B) x^(2)+x-1=0 C) x^(2)-x+1=0 D) x^(2)+x+1=0.

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