Question

Find the roots of the equation $2{x^2} + x - 6 = 0$ by factorization.

Answer 1

Hint: For solving this question we need to know the following formula: -
Factorization method is applied by splitting the middle term in which we will split the middle term such that the sum or the difference is equal to the product of the constant term and coefficient of ${x^2}.$

Complete step-by-step answer:
The given equation is $2{x^2} + x - 6 = 0$
Now, solving the equation we get
$2{x^2} + x - 6 = 0$
We get 4 & 3 as middle terms such that the sum or the difference is equal to 1 and the product of the constant term and coefficient of ${x^2}$is$ - 12$ .
$2{x^2} + 4x - 3x - 6 = 0$
We take common terms from first two terms and last two terms
$2x(x + 2) - 3(x + 2) = 0$
Further Taking common term$(x + 2)$ we get
$(2x - 3)(x + 2) = 0$
$2x - 3 = 0\,{\text{or}}\,x + 2 = 0$
Further Solving
$x = \dfrac{3}{2}\,{\text{or}}\,x = - 2$
Thus, the roots of the equation$2{x^2} + x - 6 = 0$are$\dfrac{3}{2}\,{\text{and}}\, - 2.$

Note: Now in such a case is advisable to choose the root and its sign properly. In the rough calculation all the factors of the product of the constant term and coefficient of ${x^2}.$ and then split the term according to the middle term.