Question

# Find roots of.$2{x^2} + {\text{ }}x{\text{ }}-{\text{ }}4{\text{ }} = {\text{ }}0$.

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Hint: Roots are also called x-intercepts or zeros. A quadratic function is graphically represented by a parabola with vertex located at the origin below the x-Axis or above the x-Axis.
Therefore, a quadratic function may have one, two, or zero roots when we are asked to solve a quadratic equation, we are really being asked to find the roots.
We have seen that completing the square is a useful method to solve the quadratic equation.
This method can be used to drive quadratic formulae.
$F\left( x \right) = {\text{ }}a{x^2} + {\text{ }}bx{\text{ }} + c$
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
This formula is called quadratic formula we call ${b^2} - 4ac$ is discriminant. It means if.
${b^2} - 4ac$<
${b^2} - 4ac$$= 0$
${b^2} - 4ac$$> 0$ then two real roots.

Complete step by step answer:

To find the roots of the following equation there are ‘n’ no of the method.
But we use the most used is complete the square distributing the 1 degree.
$2{x^2} + x - 4 = 0$
Firstly we multiply the coefficients of $2$ degree and then subtracting the LCM of that multiplied term such that it can form them coefficient of $1$ degree since we have no such that form is being formed.
So we now use another method i.e. quadratic formulae
$d = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Since a is 2-degree coefficient of x b is 1-degree coefficient of x
b is 1-degree coefficient of x, c is 0-degree coefficient of x
a = 2, b = 1, c = 4.
Now,
$d = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
$d = \dfrac{{ - 1 \pm \sqrt {1 + 32} }}{\begin{gathered} 4 \\ \\ \end{gathered} }$
$d = \dfrac{{ - 1 \pm \sqrt {33} }}{\begin{gathered} 4 \\ \\ \end{gathered} }$
$d = \dfrac{{ - 1 + \sqrt {33} }}{\begin{gathered} 4 \\ \\ \end{gathered} }$, $d = \dfrac{{ - 1 - \sqrt {33} }}{\begin{gathered} 4 \\ \\ \end{gathered} }$
Roots of equation $2{x^2} + x - 4 = 0$
Are ${d_1} = \dfrac{{ - 1 + \sqrt {33} }}{\begin{gathered} 4 \\ \\ \end{gathered} },{d_2} = \dfrac{{ - 1 - \sqrt {33} }}{\begin{gathered} 4 \\ \\ \end{gathered} }$

Note: Many physical and mathematical problems are in the form of quadratic equations in mathematics. The solution of the quadratic equation is of particular importance. As already discussed, a quadratic equation has no real solution if ${b^2} - 4ac < 0$. This case, as you will see in later class, is of prime importance because it helps to develop different field math in data analytics.