Question

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

Answer 1

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.