Question

How do you simplify \[\dfrac{{2{x^2} - 2}}{{{x^2} - 6x - 7}} \cdot \left( {{x^2} - 10x + 21} \right)\] ?

Answer 1

Hint: To solve the given equation, we can see that the terms are in the form of quadratic equation, hence first we need to find the factors of numerator and denominator terms then find the factors of its product term then combine all the like terms by using any of the elementary arithmetic functions i.e., addition, subtraction, multiplication and division hence simplify the terms to get the answer.

Complete step by step solution:
Let us write the given equation:
 \[\dfrac{{2{x^2} - 2}}{{{x^2} - 6x - 7}} \cdot \left( {{x^2} - 10x + 21} \right)\]
To simplify the given equation, let us first simplify
 \[\dfrac{{2{x^2} - 2}}{{{x^2} - 6x - 7}}\]
As the numerator terms consists of like terms, we get
 \[2{x^2} - 2 = 2 \cdot \left( {{x^2} - 1} \right)\]
Let us find the factorization of the obtained equation i.e., \[2 \cdot \left( {{x^2} - 1} \right)\]
We know that A difference of two perfect squares, \[{A^2} - {B^2}\] can be factored into \[\left( {A + B} \right) \cdot \left( {A - B} \right)\] , hence we get the factorization as:
 \[ \Rightarrow \] \[2 \cdot \left( {x + 1} \right) \cdot \left( {x - 1} \right)\] ………… 1
Now let us find the factors of denominator term i.e., \[{x^2} - 6x - 7\] , we can see that it is a quadratic equation of the form \[a{x^2} + bx + c\] .
The first term is, \[{x^2}\] its coefficient is 1, the middle term is, \[ - 6x\] its coefficient is -6 and the last term, the constant, is -7.
Find two factors of -7 whose sum equals the coefficient of the middle term, which is -6 i.e.,
 \[{x^2} - 6x - 7\]
 \[ \Rightarrow \] \[{x^2} - 7x + 1x - 7\]
Add up the first 2 terms, pulling out like factors:
  \[ \Rightarrow \] \[x \cdot \left( {x - 7} \right)\]
Add up the last 2 terms, pulling out common factors:
 \[ \Rightarrow \] \[1 \cdot \left( {x - 7} \right)\]
Therefore, the factors are:
 \[\left( {x + 1} \right)\left( {x - 7} \right)\] …………. 2
Now let us substitute the factors obtained i.e., equation 1 and 2 in the given equation:
 \[\dfrac{{2{x^2} - 2}}{{{x^2} - 6x - 7}} \cdot \left( {{x^2} - 10x + 21} \right)\]
 \[ \Rightarrow \] \[\dfrac{{2 \cdot \left( {x + 1} \right) \cdot \left( {x - 1} \right)}}{{\left( {x + 1} \right)\left( {x - 7} \right)}} \cdot \left( {{x^2} - 10x + 21} \right)\]
As we can see that \[\left( {x + 1} \right)\] are common, hence we get:
 \[ \Rightarrow \] \[\dfrac{{2 \cdot \left( {x - 1} \right)}}{{\left( {x - 7} \right)}} \cdot \left( {{x^2} - 10x + 21} \right)\] ………….. 3
Now let us factorize \[\left( {{x^2} - 10x + 21} \right)\] , and find the factors
Find two factors of 21 whose sum equals the coefficient of the middle term, which is -10 i.e.,
 \[{x^2} - 10x + 21\]
 \[ \Rightarrow \] \[{x^2} - 7x - 3x - 21\]
Add up the first 2 terms, pulling out like factors:
  \[ \Rightarrow \] \[x \cdot \left( {x - 7} \right)\]
Add up the last 2 terms, pulling out common factors:
 \[ \Rightarrow \] \[3 \cdot \left( {x - 7} \right)\]
Therefore, the factors are:
 \[\left( {x - 3} \right)\left( {x - 7} \right)\] ……………… 4
Now, substitute equation 4 in equation 3 as:
 \[\dfrac{{2 \cdot \left( {x - 1} \right)}}{{\left( {x - 7} \right)}} \cdot \left( {{x^2} - 10x + 21} \right)\]
 \[ \Rightarrow \] \[\dfrac{{2 \cdot \left( {x - 1} \right)}}{{\left( {x - 7} \right)}} \cdot \left( {x - 3} \right)\left( {x - 7} \right)\]
As we can see that \[\left( {x - 7} \right)\] are common, hence we get:
 \[ \Rightarrow \] \[2 \cdot \left( {x - 1} \right) \cdot \left( {x - 3} \right)\]
Therefore,
 \[\dfrac{{2{x^2} - 2}}{{{x^2} - 6x - 7}} \cdot \left( {{x^2} - 10x + 21} \right)\] = \[2 \cdot \left( {x - 1} \right) \cdot \left( {x - 3} \right)\] .
So, the correct answer is “ \[2 \cdot \left( {x - 1} \right) \cdot \left( {x - 3} \right)\] ”.

Note: The key point to simplify the given equation is that we must note, the given equation is of what type i.e., quadratic, linear, simultaneous etc, hence based on this we can solve this kind of equation. Here while solving, as it is a quadratic equation of the form \[a{x^2} + bx + c\] , we must find the factors and compare the like terms with respect to the numerator and denominator terms, then combine with its product terms and then simplify all the terms.