Question

Solve the equation: $2{{x}^{2}}+x+1=0$

Answer 1

In the question we are given a quadratic equation. To understand this question properly first we will learn what is a quadratic equation and the roots of an equation. Then we will learn the method of the discriminant formula and we will use this to solve our given equation and to get our required answer.

Complete step-by-step solution:
In the question we are given a quadratic equation.
So first we will learn what is quadratic equation and roots of equation
Quadratic equation:
It is a mathematical statement in which the degree of the variable is $2$ .
Example: ${{x}^{2}}+5$
Here the variable is $x$ and the degree of $x$ is $2$ .
Root of an equation:
If we equate an equation to $0$ , after solving it the value of the variable we get is known as the root of an equation.
Now let us learn discriminant formula for solving quadratic equation
If our given equation is $a{{x}^{2}}+bx+c=0$
Where $a$ is coefficient of ${{x}^{2}}$ and $b$ is coefficient of $x$ and $c$ is the constant term
First we will calculate discriminant$D$ ,
$D={{b}^{2}}-4ac$
Nature of roots is dependent on the value of$D$.
This will give us three cases.
Case$1$ : If the value of $D$is positive then we will get two different roots.
Case$2$ : If the value of $D$is $0$ then we will get repeated roots.
Case$3$ : If the value of $D$is negative then no real roots are possible and complex roots will exist in this case.
Then value of $x$ is calculated by
$x=\dfrac{-b\pm \sqrt{D}}{2a}$
This method is easy to solve as this involves less calculations.
Now we will proceed to our question.
In the question we are given a quadratic equation,
$2{{x}^{2}}+x+1=0$
We will solve this quadratic equation using the Discriminant formula.
First we will calculate $D$
As we know $D={{b}^{2}}-4ac$
In our given equation
$\begin{aligned} & a=2 \\\ & b=1 \\\ & c=1 \\\ \end{aligned}$
So value of $D$ is
$\begin{aligned} & D={{(1)}^{2}}-4\times 2\times 1 \\\ & = 1-8 \\\ &=-7 \\\ \end{aligned}$
Now we will find value of $x$ using value of $D$
$\begin{aligned} &\Rightarrow x=\dfrac{-1\pm \sqrt{-7}}{2\times 2} \\\ &\Rightarrow x=\dfrac{-1\pm \sqrt{-7}}{4} \\\ \end{aligned}$
As we can write $\sqrt{-7}$ as $\sqrt{7}i$
So value of $x$ is
$x=\dfrac{-1\pm \sqrt{7}i}{4}$
From this we will get two values $x$ that are
$x=\dfrac{-1+\sqrt{7}i}{4}$ And $x=\dfrac{-1-\sqrt{7}i}{4}$
Hence these are our required values of $x$.

Note: A quadratic equation can have a maximum of two roots. There is a possibility that a quadratic equation has only one root or does not have any root. We can solve quadratic equations using three methods named as factorisation or middle term splitting, discriminant method and completing the square.