Question

$sqrt{3}x^{2}+5x + 2sqrt{3}=0$

Explanation:

Step1: Combine like terms

The equation is $\sqrt{3}x^{2}+5x + 2\sqrt{3}=0$. For a quadratic equation $ax^{2}+bx + c = 0$, here $a=\sqrt{3}$, $b = 5$, $c=2\sqrt{3}$.

Step2: Use quadratic formula

The quadratic formula is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. First, calculate the discriminant $\Delta=b^{2}-4ac$.
Substitute the values: $\Delta = 5^{2}-4\times\sqrt{3}\times2\sqrt{3}=25 - 8\times3=25 - 24 = 1$.

Step3: Solve for x

Now, $x=\frac{-5\pm\sqrt{1}}{2\sqrt{3}}=\frac{-5\pm1}{2\sqrt{3}}$.
For the plus sign: $x_1=\frac{-5 + 1}{2\sqrt{3}}=\frac{-4}{2\sqrt{3}}=\frac{-2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}$.
For the minus sign: $x_2=\frac{-5 - 1}{2\sqrt{3}}=\frac{-6}{2\sqrt{3}}=\frac{-3}{\sqrt{3}}=-\sqrt{3}$.

Answer:

The solutions are $x = -\frac{2\sqrt{3}}{3}$ and $x=-\sqrt{3}$