How do you solve $(2 + \sqrt{3}) x^{2} + (3 + \sqrt{3}) x + 2 = 0$ $(2+sqrt(3))x^2+(3+sqrt(3))x+2 = 0$ ?

Question

How do you solve $(2 + \sqrt{3}) x^{2} + (3 + \sqrt{3}) x + 2 = 0$ $(2+sqrt(3))x^2+(3+sqrt(3))x+2 = 0$ ?

1 Answer

Impact of this question

1436 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-03-26T13:50:52

$x = \frac{- 3 + \sqrt{3} \pm i (\sqrt{3} - 1)}{2}$ $x = (-3+sqrt(3)+-i (sqrt(3)-1))/2$

Explanation:

$(2 + \sqrt{3}) x^{2} + (3 + \sqrt{3}) x + 2 = 0$ $(2+sqrt(3))x^2+(3+sqrt(3))x+2 = 0$

is in the standard form:

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

with $a = 2 + \sqrt{3}$ $a=2+sqrt(3)$ , $b = 3 + \sqrt{3}$ $b = 3+sqrt(3)$ and $c = 2$ $c=2$

Let's look at the discriminant first...

$Delta = b^2-4ac$

$color(white)(Delta) = (3+sqrt(3))^2-4(2+sqrt(3))(2)$

$color(white)(Delta) = 9+6sqrt(3)+3-16-8sqrt(3)$

$color(white)(Delta) = -4-2sqrt(3)$

Since $Delta < 0$ this quadratic has no real solutions.

We can still use the quadratic formula to find the complex solutions:

$x = (-b+-sqrt(b^2-4ac))/(2a)$

$color(white)(x) = (-b+-sqrt(Delta))/(2a)$

$color(white)(x) = (-3-sqrt(3)+-sqrt(-4-2sqrt(3)))/(4+2sqrt(3))$

$color(white)(x) = (-3-sqrt(3)+-i sqrt(4+2sqrt(3)))/(4+2sqrt(3))$

$color(white)(x) = ((-3-sqrt(3)+-i sqrt(4+2sqrt(3)))(2-sqrt(3)))/((4+2sqrt(3))(2-sqrt(3)))$

$color(white)(x) = ((-3-sqrt(3)+-i sqrt(2(2+sqrt(3))))(2-sqrt(3)))/2$

$color(white)(x) = (-3+sqrt(3)+-i sqrt(2(2-sqrt(3))))/2$

Note that:

$(sqrt(3)-1)^2 = sqrt(3)^2-2sqrt(3)+1 = 4-2sqrt(3)$

So:

$sqrt(2(2-sqrt(3))) = sqrt(3)-1$

So:

$x = (-3+sqrt(3)+-i (sqrt(3)-1))/2$

Science

Math

Humanities

... and beyond

How do you solve $(2 + \sqrt{3}) x^{2} + (3 + \sqrt{3}) x + 2 = 0$ $(2+sqrt(3))x^2+(3+sqrt(3))x+2 = 0$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you solve (2+sqrt(3))x^2+(3+sqrt(3))x+2 = 0(2+√3)x2+(3+√3)x+2=0(2+sqrt(3))x^2+(3+sqrt(3))x+2 = 0 ?

1 Answer

Explanation:

Related questions

Impact of this question

How do you solve $(2 + \sqrt{3}) x^{2} + (3 + \sqrt{3}) x + 2 = 0$ $(2+sqrt(3))x^2+(3+sqrt(3))x+2 = 0$ ?