Question

How do you solve $ {{x}^{2}}+4x+2=0 $ using the quadratic formulae?

Answer 1

Hint: For answering this question we will use the concept of quadratic equations and use the formulae for finding the solutions of the quadratic equation $ a{{x}^{2}}+bx+c=0 $ given as $ \dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} $ and reducing and simplifying it using some algebraic concepts.

Complete step by step answer:
Now considering from the question we have an equation $ {{x}^{2}}+4x+2=0 $ we need to find the solutions of this equation.
From the basic concepts of algebra and quadratic equations we know that the solutions of any quadratic equation $ a{{x}^{2}}+bx+c=0 $ is given by the formulae $ \dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} $ .
By comparing the given equation $ {{x}^{2}}+4x+2=0 $ with $ a{{x}^{2}}+bx+c=0 $ we can conclude that $ a=1 $ , $ b=4 $ and $ c=2 $ .
By using these values we will have the solutions of the equation as $ \dfrac{-\left( 4 \right)\pm \sqrt{{{\left( 4 \right)}^{2}}-4\left( 1 \right)\left( 2 \right)}}{2\left( 1 \right)} $ .
By further simplifying this we will have $ \dfrac{-4\pm \sqrt{16-8}}{2}=\dfrac{-4\pm \sqrt{8}}{2}=-2\pm \sqrt{2} $ .
Therefore we can conclude that the solutions of the given quadratic equation $ {{x}^{2}}+4x+2=0 $ is $ -2\pm \sqrt{2} $ .

Note:
  We should be sure with the calculations and the concept that we are going to apply in questions of this type. This question can be answered in another way also but as in the question it is asked to do it using quadratic formula so we have to prefer the above-discussed method. But the other method is by reducing the equations that is the given quadratic equation $ {{x}^{2}}+4x+2=0 $ is reduced to $ {{\left( x+2 \right)}^{2}}-2=0 $ hence the solutions are given by $ {{\left( x+2 \right)}^{2}}=2 $ and $ x+2=\pm \sqrt{2}\Rightarrow x=-2\pm \sqrt{2} $ . The solutions for the given equations are the same obtained by both methods.