Question

How do you factor $6{x^2} - 13x - 28$?

Answer 1

Hint:We will first mention the general quadratic equation and then mention the formula of the roots and then putting the values as compared to the given equation, we get the answer.

Complete step by step answer:We are given that we are required to factor $6{x^2} - 13x - 28$ .
The general quadratic equation is given by $a{x^2} + bx + c = 0$, where a, b and c are the real numbers.
The roots of this general quadratic equation is given by the following formula:-
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Now, if we compare the given quadratic equation $6{x^2} - 13x - 28$ to the general quadratic equation $a{x^2} + bx + c = 0$, we will then obtain the following:-
$ \Rightarrow $a = 6, b = - 13 and c = 28
Now, putting these values in the formula of quadratics, we will then obtain the following:-
$ \Rightarrow x = \dfrac{{ - ( - 13) \pm \sqrt {{{( - 13)}^2} - 4(6)( - 28)} }}{{2(6)}}$
Simplifying some calculations in the above equation, we will then obtain the following equation:-
$ \Rightarrow x = \dfrac{{13 \pm \sqrt {169 + 672} }}{{12}}$
Simplifying the calculations inside the square – root in the above equation, we will then obtain the following equation:-
$ \Rightarrow x = \dfrac{{13 \pm \sqrt {841} }}{{12}}$
Simplifying the calculations inside the square – root further in the above equation, we will then obtain the following equation:-
$ \Rightarrow x = \dfrac{{13 \pm 29}}{{12}}$
Therefore, the values of x can be $\dfrac{7}{2}$ and $ - \dfrac{4}{3}$.
Hence, the equation can be written as $\left( {x - \dfrac{7}{2}} \right)\left( {x + \dfrac{4}{3}} \right) = 0$.
Thus the equation is: $\left( {2x - 7} \right)\left( {3x + 4} \right) = 0$

Note:
The students must note that, if not mentioned that we are required to solve it by any particular method, then we may also use the method of splitting the middle term like the following:-
We are given that we are required to solve $6{x^2} - 13x - 28$ .
We can write the given equation as:-
$ \Rightarrow 6{x^2} - 21x + 8x - 28 = 0$
Taking 3x common from first two terms, we will then obtain the following equation:-
$ \Rightarrow 3x(2x - 7) + 8x - 28 = 0$
Taking 4 common from first the last two terms, we will then obtain the following equation:-
$ \Rightarrow 3x(2x - 7) + 4(2x - 7) = 0$
Now taking (2x – 7) common, we will then obtain the following:-
$ \Rightarrow (2x - 7)(3x + 4) = 0$