Question

What are the solutions to $3{{x}^{2}}+12x+6=0$?
A)$-2\pm \sqrt{2}$
B)$-2\pm \dfrac{\sqrt{30}}{3}$
C)$-6\pm \sqrt{2}$
D)$-6\pm 6\sqrt{2}$

Answer 1

To solve this question we should have the knowledge of quadratic equations. We should know the methods of finding the roots of the quadratic equation. We should have the knowledge of checking the nature of roots. The solutions of the quadratic can also be found out with the help of graphs. In the above question we should know the formula for finding roots of quadratic equations and substitute the values accordingly.

Complete step-by-step solution:
The given question requires the concept of quadratic equations. We have to know how to find the roots of the given quadratic equation. The natural roots can be identified by a formula which can give us the idea that in which category of numbers the required roots lie. It requires simple substitution of the values from the quadratic equation by understanding the concepts. Let us now discuss the above concepts.
Quadratic Equation: The algebraic equation consisting of just one variable with index (highest power) as $2$ is known as a quadratic equation. In a standard quadratic equation the right hand side of the equation is always $0$.
$a{{x}^{2}}+bx+c=0$ is the standard format of a quadratic equation.
For e.g. Take a quadratic equation
$2{{x}^{2}}-8x+3=0$
Here, $a=2$, $b=-8$ and $c=3$
Roots of a quadratic equation: The value of the variable present in the quadratic equation which satisfies the given quadratic equation is called as the root of the given quadratic equation. A quadratic equation has at least one and at most two roots. The roots can be real and equal, real and unequal, imaginary according to the value of discriminant.
Considering the standard format of quadratic equation, the formula for finding the roots of quadratic equation is given by,
$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
After revising all the concepts required let us now take a look at the main question.
We have been given a quadratic equation $3{{x}^{2}}+12x+6=0$
The given quadratic equation can also be written as ${{x}^{2}}+4x+2=0$
Here, $a=1$, $b=4$, $c=2$
By substituting the values in the above formula we get,
$x=\dfrac{-4\pm \sqrt{{{(4)}^{2}}-4\times 1\times 2}}{2\times 1}$

& \Rightarrow x=\dfrac{-4\pm \sqrt{16-8}}{2} \\\ & \Rightarrow x=\dfrac{-4\pm \sqrt{8}}{2} \\\ & \Rightarrow x=\dfrac{-4\pm 2\sqrt{2}}{2} \\\ & \Rightarrow x=-2\pm \sqrt{2} \\\ \end{aligned}$$ From this we can conclude that option A is the correct answer. **No need to check other options as we have got our definite answer.** **Note:** This question was totally based on the concepts of quadratic equations and its roots. The checking of multiple options is not required in this question. We can also find the answer by checking all options but it will take a huge amount of time as all the options contain rational numbers and it will just make the calculations lengthy. Hence, finding the roots by conventional method is the best approach for this question.