Question

Solve $ \dfrac{{10{x^2} + 15x + 63}}{{5{x^2} - 25x + 12}} = \dfrac{{2x + 3}}{{x - 5}} $

Answer 1

Hint: Here we will cross multiply and will simplify the equation making the pair of like terms and then finding the required value for “x”. Here the product of one quadratic equation with the linear equation happens.

Complete step-by-step answer:
Do cross multiplication. Where the numerator of one side is multiplied with the denominator and the denominator is multiplied with the numerator of the opposite side.
 $ \Rightarrow (10{x^2} + 15x + 63)(x - 5) = (2x + 3)(5{x^2} - 25x + 12) $
Open the brackets on both the sides of the equation. When you multiply one positive term with one negative term then resultant value is negative, when you multiply both the negative terms then the resultant value is positive.
 $ \Rightarrow 10{x^3} - 50{x^2} + 15{x^2} - 75x + 63x - 315 = 10{x^3} - 50{x^2} + 24x + 15{x^2} - 75x + 36 $
Take all the terms on the left hand side of the equation. When you move any term from one side to another, the sign also changes. Positive terms become negative and vice-versa.
 $ \Rightarrow 10{x^3} - 50{x^2} + 15{x^2} - 75x + 63x - 315 - 10{x^3} + 50{x^2} - 24x - 15{x^2} + 75x - 36 = 0 $
Make the pair of like terms in the above equation –
\[ \Rightarrow \underline {10{x^3} - 10{x^3}} - \underline {50{x^2} + 50{x^2} + 15{x^2} - 15{x^2}} - \underline {75x + 63x - 24x + 75x} - \underline {315 - 36} = 0\]
Like terms with equal values and opposite sign cancel each other. Simplify the above equation-
 $ \Rightarrow 39x - 351 = 0 $
Make the variable “x” the subject –
 $ \Rightarrow 39x = 351 $
When the term multiplicative on one end is moved to the opposite side then it goes in the denominator.
 $ \Rightarrow x = \dfrac{{351}}{{39}} $
Find the factors of the numerator on the right hand side of the equation –
 $ \Rightarrow x = \dfrac{{39 \times 9}}{{39}} $
Common factors from the numerator and the denominator cancel each other. Therefore remove from the numerator and the denominator.
 $ \Rightarrow x = 9 $
So, the correct answer is “x=9”.

Note: Always remember when you move any term from one side to another, the sign of the term also changes. Positive terms become negative and negative terms become positive. Be careful about the sign.
While doing simplification remember the golden rules-
I.Addition of two positive terms gives the positive term
II.Addition of one negative and positive term, you have to do subtraction and give sign of bigger numbers, whether positive or negative.
III.Addition of two negative numbers gives a negative number but in actual you have to add both the numbers and give a negative sign to the resultant answer.