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

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.