Question

How do you solve the expression given as$\dfrac{{{x^2} - 7x + 10}}{{{x^2} - 10x + 25}} \times \dfrac{{x - 5}}{{x - 2}}$?

Answer 1

For such expression you have to solve by the system of eliminating the common factors from the equation, to obtain common factor you have to simplify the first fraction by mid term splitting rule, where you can have the factors of both the numerator and denominator, and the factors which got cancelled out are needed to be cancelled.

Complete step by step solution:
The given expression is $\dfrac{{{x^2} - 7x + 10}}{{{x^2} - 10x + 25}} \times \dfrac{{x - 5}}{{x - 2}}$

First we have to find the factors of the first fraction be mid term splitting rule, on solving we get:

$$\Rightarrow {x^2} - 7x + 10 = {x^2} - (5 + 2)x + 10 = {x^2} - 5x - 2x + 10 = x(x - 5) - 2(x - 5) = (x - 5)(x - 2) \\\ \Rightarrow {x^2} - 10x + 25 = {x^2} - (5 + 5)x + 25 = {x^2} - 5x - 5x + 25 = x(x - 5) - 5(x - 5) = (x - 5)(x - 5) \\\$$

Here we obtained the possible factors of both the numerator and denominator of the first fraction, now putting the factors in the main expression we get:

$$\Rightarrow \dfrac{{(x - 5)(x - 2)}}{{(x - 5)(x - 5)}} \times \dfrac{{x - 5}}{{x - 2}} \\\ \Rightarrow 1 \\\$$

Here all the terms got cancelled out between the two fractions since the number of terms in numerator the denominator are the same and equal and the result of the expression is “1”.

Additional Information: In dividing the term by the HCF of both terms you should take care that both the terms are being divided every time you use the division, and the final outcome of any fraction is the least fraction that can be obtained.

Note: The fraction terms are obtained when the last possible division without making to decimal is completed, now the lowest fraction term can be written as in fraction or you can also write in decimal terms after reduction of the last smallest fraction, fraction can be of two type rational and irrational.