Question

Find the value of x from the given equation, \[\left( {3x - 8} \right)\left( {3x + 2} \right) - \left( {4x - 11} \right)\left( {2x + 1} \right) = \left( {x - 3} \right)\left( {x + 7} \right)\]

Answer 1

Hint: Here we will multiply and simplify the equation and then find the value of x from the equation so formed.

Complete step-by-step answer:
The given equation is:-
\[\left( {3x - 8} \right)\left( {3x + 2} \right) - \left( {4x - 11} \right)\left( {2x + 1} \right) = \left( {x - 3} \right)\left( {x + 7} \right)\]
Multiplying and simplifying the equation we get:-
\[\left( {3x} \right)\left( {3x} \right) + 2\left( {3x} \right) - 8\left( {3x} \right) - \left( 8 \right)\left( 2 \right) - \left[ {\left( {4x} \right)\left( {2x} \right) + 1\left( {4x} \right) - 11\left( {2x} \right) - 11\left( 1 \right)} \right] = x\left( x \right) + 7\left( x \right) - 3\left( x \right) - \left( 3 \right)\left( 7 \right)\]
Simplifying it further we get:-
\[9{x^2} + 6x - 24x - 16 - \left[ {8{x^2} + 4x - 22x - 11} \right] = {x^2} + 7x - 3x - 21\]
Solving it further we get:-
\[9{x^2} - 18x - 16 - 8{x^2} + 18x + 11 = {x^2} + 4x - 21\]
Simplifying it we get:-
\[{x^2} - 5 = {x^2} + 4x - 21\]
Cancelling the required terms we get:-
\[4x = 21 - 5\]
Solving for x we get:-
\[4x = 16\]
\[x = \dfrac{{16}}{4}\]
Simplifying it we get:-
\[x = 4\]

Hence the value of x is 4.

Note: In such problems, we first need to rearrange the given equation and then simplify it and solve it for the value of x.
If it is linear i.e. the highest power of the variable is one then we can get the value directly by performing ordinary operations like addition, subtraction, multiplication and division.
While if the equation turns out to be quadratic then we need to use a middle term split or quadratic formula to solve for the value of the variable.