Question

Solve the following:
 \[{{x}^{2}}+\dfrac{x}{\sqrt{2}}+1=0\]

Answer 1

Hint: In this question we firstly we will check that the given equation is quadratic equation as it has the degree of two and then we will compare the given equation from general quadratic equation, then we will find out the discriminant after that we will apply the quadratic formula so that we can obtain the result.

Formula used:
\[\Rightarrow x=\dfrac{-b\pm \sqrt{D}}{2a}\]
Where \[D={{b}^{2}}-4ac\]

Complete step by step answer:
Now according to the question we have given a quadratic equation that is \[{{x}^{2}}+\dfrac{x}{\sqrt{2}}+1=0\]
Compare this given quadratic equation by general equation \[a{{x}^{2}}+bx+c=0\] so that we can get the values of \[a,b\] and \[c\]
\[\Rightarrow a=1,b=\dfrac{1}{\sqrt{2}},c=1\]
Discriminant can be find out by the following formula:
\[\Rightarrow D={{b}^{2}}-4ac\]
Put the values of \[a,b\] and \[c\]
\[\Rightarrow D={{\left( \dfrac{1}{\sqrt{2}} \right)}^{2}}-4\times 1\times 1\]
\[\Rightarrow D=\dfrac{1}{2}-4\]
Take the LCM to simplify it:
\[\Rightarrow D=\dfrac{1-8}{2}\]
\[\Rightarrow D=\dfrac{-7}{2}\]
Here the discriminant is less than zero and we know that if \[D<0\] then the roots of the quadratic equation will be complex.
Now apply the quadratic formula:
\[\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
Where \[D={{b}^{2}}-4ac\]
\[\Rightarrow x=\dfrac{-b\pm \sqrt{D}}{2a}\]
 Put the values of \[a,b\] and \[c\] we get:
\[\Rightarrow x=\dfrac{-\dfrac{1}{\sqrt{2}}\pm \sqrt{\dfrac{-7}{2}}}{2\times 1}\]
\[\Rightarrow x=\dfrac{-\dfrac{1}{\sqrt{2}}\pm \sqrt{\dfrac{-7}{2}}}{2}\]
Split the terms we get:
\[\Rightarrow x=\dfrac{-\dfrac{1}{\sqrt{2}}}{2}\pm \dfrac{\sqrt{\dfrac{-7}{2}}}{2}\]
\[\Rightarrow x=\dfrac{-1}{2\sqrt{2}}\pm \dfrac{1}{2}\cdot \sqrt{\dfrac{-7}{2}}\]
\[\Rightarrow x=\dfrac{-1}{2\sqrt{2}}\pm \dfrac{1}{2}\cdot \sqrt{\dfrac{7}{2}\times -1}\]
We know that \[i=\sqrt{-1}\]
\[\Rightarrow x=\dfrac{-1}{2\sqrt{2}}\pm \dfrac{\sqrt{7}}{2\sqrt{2}}\cdot i\]

Note:
The term \[{{b}^{2}}-4ac\] in the quadratic formula is known as the discriminant of a quadratic equation. The discriminant of a quadratic equation reveals the nature of roots.
We can solve quadratic equations using the completing square method and factorisation method as well.