Question

How do you solve $2{x^2} + 4x - 1 = 7{x^2} - 7x + 1$?

Answer 1

Hint:
We will first bring all the terms from the right hand side to the left hand side. Then, we will combine the like terms to form a quadratic equation and thus, we will use the quadratics formula to solve.

Complete step by step solution:
We are given that we are required to solve $2{x^2} + 4x - 1 = 7{x^2} - 7x + 1$.
Taking all the terms from the right hand side to left hand side, we will then obtain the following equation with us:-
\[ \Rightarrow \left\{ {2{x^2} + 4x - 1} \right\} - \left\{ {7{x^2} - 7x + 1} \right\} = 0\]
Removing the parenthesis and then writing the same equation, we will get the following equation with us:-
$ \Rightarrow 2{x^2} + 4x - 1 - 7{x^2} + 7x - 1 = 0$
Simplifying the left hand side of the above equation by clubbing the like terms, we will then obtain the following equation with us:-
$ \Rightarrow - 5{x^2} + 11x - 2 = 0$
We can write the middle term as follows:-
$ \Rightarrow - 5{x^2} + 10x + x - 2 = 0$
Taking $ - 5x$ common from the first two terms on the left had side of the above equation, we will then obtain the following equation with us:-
$ \Rightarrow - 5x\left( {x - 2} \right) + x - 2 = 0$
Taking 1 common out of the last two terms, we will then obtain the following equation with us:-
$ \Rightarrow - 5x\left( {x - 2} \right) + \left( {x - 2} \right) = 0$
Taking $\left( {x - 2} \right)$ common out of it, we will then obtain the following equation with us:-
$ \Rightarrow \left( {x - 2} \right)\left( { - 5x + 1} \right) = 0$
Thus, we have the possible values of $x$ as 2 and $\dfrac{1}{5}$.

Note:
The students must note that we have an alternate way of solving the same equation. We will use the quadratic formula which states that:
The general quadratic equation is given by $a{x^2} + bx + c = 0$, where a, b and c are the constants.
The roots are this equation is given by the following expression with us:-
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Comparing the general quadratic equation, we have: a = - 5, b = 11 and c = - 2.
Therefore, the roots of the equation $ - 5{x^2} + 11x - 2 = 0$ are given by:-
$ \Rightarrow x = \dfrac{{ - 11 \pm \sqrt {{{11}^2} - 4( - 5)( - 2)} }}{{2( - 5)}}$
Simplifying the calculations, we get the following equation with us:-
$ \Rightarrow x = \dfrac{{ - 11 \pm \sqrt {121 - 40} }}{{ - 10}}$
Simplifying the calculations further, we get the following equation with us:-
$ \Rightarrow x = \dfrac{{ - 11 \pm 9}}{{ - 10}}$
Thus, we have the possible values of x as 2 and $\dfrac{1}{5}$.