Question

solve √3x2 +x+√3=0 from complex numbers and quadratic equation ​

Answer 1

$\sf\huge\bold{Answer:}$

Given :

$\sf:⟶ \sqrt{3} {x}^{2} + x + \sqrt{3} = 0$

To find :

Value of x

Solution :

$\fbox\purple{Solving Quadratic Equation Using discriminant method }$

① Getting values of a, b and c by comparing given equation with general form.

$\sf:⟶ \sqrt{3} {x}^{2} + x + \sqrt{3} = 0$

$\sf:⟶ax {}^{2} + bx + c = 0$

we get,

a=√3

b=1

c=√3

② Finding discriminant

:⟶D=b²-4ac

:⟶D=1²-4×√3×√3

:⟶D=1-4×3

:⟶D=1-12

:⟶D= -11

③ Finding roots

$\sf:⟶x = \frac{ - b± \sqrt{D} }{2a}$

$\sf:⟶x = \frac{ - 1 ± \sqrt{-11} }{2×√3}$

$\sf:⟶x = \frac{ - 1± \sqrt{-1×11} }{2√3}$

√-1 = $i$

(iota) .→ This symbol is used for representation of complex numbers.

$\sf:⟶x = \frac{ - 1± \sqrt{11} i}{2√3}$

Rationalising denominator

$\sf:⟶x = \frac{ - 1± \sqrt{11}i }{2 \sqrt{3} } \times \frac{2 \sqrt{3} }{2 \sqrt{3} }$

$\sf:⟶x = \frac{2 \sqrt{3}( - 1± \sqrt{11}i )}{2 \sqrt{3}× 2 \sqrt{3} }$

$:⟶ \sf \:x = \frac{ - 2 \sqrt{3}±( 2) \sqrt{11} \times \sqrt{3}i }{2 \times 2 \times \sqrt{3 } \times \sqrt{3} }$

$\sf:⟶x = \frac{ - 2 \sqrt{3} ±2 \sqrt{33}i }{4 \times 3}$

$\sf:⟶x = \frac{ - 2 \sqrt{3} ±2 \sqrt{33}i }{12}$

$\sf:⟶x = \frac{ 2 (-\sqrt{3} ± \sqrt{33}i) }{2 \times 6}$

$\sf:⟶x = \frac{ -\sqrt{3} ± \sqrt{33}i}{ 6}$

$\sf\huge{x = \frac{ -\sqrt{3} ± \sqrt{33}i}{ 6} }$

or

$\huge\purple{x = \frac{ -\sqrt{3} + \sqrt{33}i}{ 6} }$

and

$\huge\purple{x = \frac{ -\sqrt{3} - \sqrt{33}i}{ 6}}$

Answer 2

Answer:

$\mathbf{x = \dfrac{-1 \pm i\sqrt{11}}{2\sqrt3}}$

Step-by-step explanation:

Given :

$\mathrm{\sqrt3x^2 + x + \sqrt3}$

To find :

Solution of the given equation

Solution :

We know that if ax² + bx + c = 0 is a quadratic equation, then, Solutions of it x are :

$\boxed{\mathrm{x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}}}$

Here, a = √3 ; b = 1 ; c = √3

$\implies \mathrm{x = \dfrac{-1 \pm \sqrt{1^2 - 4(\sqrt3)(\sqrt3)}}{2\sqrt3}}$

$\implies \mathrm{x = \dfrac{-1 \pm \sqrt{1 - 4(3)}}{2\sqrt3}}$

$\implies \mathrm{x = \dfrac{-1 \pm \sqrt{-11}}{2\sqrt3}}$

$\implies \mathrm{x = \dfrac{-1 \pm i\sqrt{11}}{2\sqrt3}}$

Where, i = √-1.

So,

$\therefore \underline{\boxed{\mathrm{x = \dfrac{-1 \pm i\sqrt{11}}{2\sqrt3}}}}$

This is the solution of the given equation.

Hope it helps!!