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Solving a Quadratic Equation by Factorization Method

The numerator...

Question

The numerator of a fraction is 3 less than the denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and the original fraction is $\frac{29}{20}$ . Find the original fraction.

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Solution

Let the denominator be x.
Therefore numerator=x-3
After adding 2 to both the numerator and denominator,
New numerator=x-3+2=x-1
New denominator=x+2.

According to question,
$fraction numerator x minus 1 over denominator x plus 2 end fraction plus fraction numerator x minus 3 over denominator x end fraction equals 29 over 20 rightwards double arrow fraction numerator x squared minus x plus x squared minus x minus 6 over denominator x squared plus 2 x end fraction equals 29 over 20 rightwards double arrow 40 x squared minus 40 x minus 120 equals 29 x squared plus 58 x rightwards double arrow 11 x squared minus 98 x minus 120 equals 0 rightwards double arrow 11 x squared minus 110 x plus 12 x minus 120 equals 0 rightwards double arrow 11 x left parenthesis x minus 10 right parenthesis plus 12 left parenthesis x minus 10 right parenthesis equals 0 rightwards double arrow left parenthesis 11 x plus 12 right parenthesis left parenthesis x minus 10 right parenthesis equals 0 rightwards double arrow x equals negative 12 over 11 space o r space 10$
Thus if
x= $fraction numerator negative 12 over denominator 11 end fraction$
Then the fraction= $fraction numerator begin display style fraction numerator negative 12 over denominator 11 end fraction minus 3 end style over denominator begin display style fraction numerator negative 12 over denominator 11 end fraction end style end fraction equals fraction numerator begin display style fraction numerator negative 12 minus 33 over denominator 11 end fraction end style over denominator begin display style fraction numerator negative 12 over denominator 11 end fraction end style end fraction equals 35 over 12 space space space left parenthesis w h i c h space d o e s n o t space s a t i s f y space t h e space e q u a t i o n right parenthesis$

If x=10
Then the fraction= $fraction numerator 10 minus 3 over denominator 10 end fraction equals 7 over 10$

Thus the original fraction is

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