Question

Find the roots of the following equation using the quadratic formula-
$2{{\text{x}}^2} + {\text{x}} - 4 = 0$

Answer 1

Hint: The roots of the equation are the values at which the value of the function becomes zero, or the graph of the functions cut the x-axis. The formula for the roots of a quadratic equation is given by-
${\text{x}} = \dfrac{{{\text{ - b}} \pm \sqrt {{{\text{b}}^2} - 4ac} }}{{2{\text{a}}}}$

Complete step-by-step answer:
We have been given the equation-
$2{{\text{x}}^2} + {\text{x}} - 4 = 0$
We will compare this with the equation, $a{x^2} + bx + {\text{c}} = 0$
By comparing we get the values of a, b and c as-
a = 2, b = 1 and c = -4
Substituting these values in the quadratic formula we get-
$\begin{align}
  &{\text{x}} = \dfrac{{ - 1 \pm \sqrt {{1^2} - 4 \times 2 \times \left( { - 4} \right)} }}{{2 \times 2}} \\
  &{\text{x}} = \dfrac{{ - 1 \pm \sqrt {1 + 32} }}{4} \\
  &{\text{x}} = \dfrac{{ - 1 + \sqrt {33} }}{4},\dfrac{{ - 1 - \sqrt {33} \;}}{4} \\
\end{align} $

These are the required roots of the equation.

Note: While using the quadratic formula, it is important to use the plus-minus sign, as it gives two roots of the equation. Also, if the quantity inside the square root, which is the discriminant, is negative, then real roots of the equation do not exist. Students often ignore the negative sign and write the answer. We can find roots of quadratic equations by factorization method as well.