Question

Solve the equation \[\dfrac{{3{x^2} + 21x + 4}}{{3{x^2} + 27x + 5}} = \dfrac{{x + 7}}{{x + 9}}\]
A. 1
B. 6
C. 5
D. 4

Answer 1

Hint: As this is equation-based questions so it must have to be changed in the form of LHS and RHS. So here it can be changed by cross-multiplying the equations on both sides. Or it can also be simplified by doing ordinary multiplication on both sides and after that solve for the variable.

Complete step-by-step answer:
The given equation is,
\[\dfrac{{3{x^2} + 21x + 4}}{{3{x^2} + 27x + 5}} = \dfrac{{x + 7}}{{x + 9}}\]
Multiplying both sides by $\left( {3{x^2} + 27x + 5} \right)\left( {x + 9} \right)$
\[ \Rightarrow \dfrac{{3{x^2} + 21x + 4}}{{3{x^2} + 27x + 5}} \times \left( {3{x^2} + 27x + 5} \right)\left( {x + 9} \right) = \dfrac{{x + 7}}{{x + 9}} \times \left( {3{x^2} + 27x + 5} \right)\left( {x + 9} \right)\]
On simplifying further,
$ \Rightarrow \left( {3{x^2} + 21x + 4} \right)\left( {x + 9} \right) = \left( {3{x^2} + 27x + 5} \right)\left( {x + 7} \right)$
$ \Rightarrow x\left( {3{x^2} + 21x + 4} \right) + 9\left( {3{x^2} + 21x + 4} \right) = x\left( {3{x^2} + 27x + 5} \right) + 7\left( {3{x^2} + 27x + 5} \right)$
On multiplying the terms in LHS and RHS,
\[ \Rightarrow 3{x^3} + 21{x^2} + 4x + 27{x^2} + 189x + 36 = 3{x^3} + 27{x^2} + 5x + 21{x^2} + 189x + 35\]
\[ \Rightarrow 3{x^3} + 48{x^2} + 193x + 36 = 3{x^3} + 48{x^2} + 194x + 35\]
On subtracting \[3{x^3} + 48{x^2} + 193x + 35\] both sides,
\[ \Rightarrow 3{x^3} + 48{x^2} + 193x + 36 - \left( {3{x^3} + 48{x^2} + 193x + 35} \right) = 3{x^3} + 48{x^2} + 194x + 35 - \left( {3{x^3} + 48{x^2} + 193x + 35} \right)\]\[ \Rightarrow 3{x^3} + 48{x^2} + 193x + 36 - 3{x^3} - 48{x^2} - 193x - 35 = 3{x^3} + 48{x^2} + 194x + 35 - 3{x^3} - 48{x^2} - 193x - 35\]
On simplifying further the equation becomes,
$ \Rightarrow 1 = x$
Or,
$x = 1$
$\therefore {\text{ x = 1 is the solution of the given equation}}{\text{.}}$
So option (A) is correct.

Note: In this type of questions the equations are simplified as far as possible and then it is solved for the variable in the equation. Calculations should be done carefully to avoid any mistake. During multiplication the power of the variable should be handled carefully otherwise it leads to the wrong solution. Multiply the coefficients of the variable carefully because if the coefficients are wrong then the final answer would be changed. If the final answer comes in fraction then it may be changed into decimal in rounded form. If the simplified equation comes in quadratic form then it can solved using quadratic formula $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ , where a, b and c are the constants in the general quadratic equation of the form $a{x^2} + bx + c = 0$ .The values of a, band c can be found out be comparing it with the simplified equation. Always try to solve step by step. After the final answer is found out it can be checked whether it satisfies the original equation given in the question by simply substituting its value in the equation.