Question

Find the roots of the quadratic equation $ 2{x^2} + x + 4. $

Answer 1

Hint: A quadratic equation is an equation in the form of $ a{x^2} + bx + c = 0, $ where a is not equals to zero. The roots of the quadratic equations are the numbers that satisfy the equation. There are always two roots for any quadratic equation. So, here after solving the quadratic equation, we will get two roots by equating the given equation equals to zero and by applying the formula.

Complete step-by-step answer:
equation.
So, at first, we will put the given equation equals to zero, i.e., $ 2{x^2} + x + 4 = 0 $
Now we will apply the ShreeDharacharya formula, which is
$\Rightarrow x = \dfrac{{ - 1 \pm \sqrt D }}{{4a}} $ , where, discriminant, $ D = {b^2} - 4ac. $
The given equation $ 2{x^2} + x + 4 = 0 $ is in the form of
$ a{x^2} + bx + c = 0, $ where values \[a{\text{ }} = {\text{ }}2,{\text{ }}b{\text{ }} = {\text{ }}1\] and \[c{\text{ }} = {\text{ }}4.\] Using these, we will first find the value of discriminant and then by putting the value of D in the formula, we will find the roots of the equation.
So, on applying formula by putting the values \[a{\text{ }} = {\text{ }}2,{\text{ }}b{\text{ }} = {\text{ }}1\] and \[c{\text{ }} = {\text{ }}4,\] we get
Discriminant,
$ \Rightarrow D = {(1)^2} - 4(2)(4) = 1 - 32 = - 31 $
Now the roots of the equation are, $ x = \dfrac{{ - 1 \pm \sqrt { - 31} }}{{2(2)}} = \dfrac{{ - 1 \pm \sqrt {31i} }}{4} $
 $\Rightarrow x = \dfrac{{ - 1 + \sqrt {31i} }}{4} $ and $ x = \dfrac{{ - 1 - \sqrt {31i} }}{4} $
Thus, the roots of the quadratic equation $ 2{x^2} + x + 4 $ are $ x = \dfrac{{ - 1 + \sqrt {31i} }}{4} $ and $ x = \dfrac{{ - 1 - \sqrt {31i} }}{4} $ .

Note: There is another method to find the roots of the quadratic equation, which is factorization method, where the middle term is break in a way that sum is equals to the coefficient of x and multiplication of those terms should be equals to coefficient of $ {x^2} $ and constant. But when this method doesn’t work Shree Dharcharya’s formula is used to find the roots, just like we did here.