\(\frac{x^2-8}{x^2-16}=\frac{1}{x+4}+\frac{1}{x-4}\)

\(\Rightarrow\frac{x^2-8}{\left(x+4\right)\left(x-4\right)}=\frac{x-4}{\left(x+4\right)\left(x-4\right)}+\frac{x+4}{\left(x-4\right)\left(x+4\right)}\)

\(\Rightarrow x^2-8=x-4+x+4\)

\(\Rightarrow x^2-8=2x\)

\(\Rightarrow x^2-2x-8=0\)

\(\Delta=b^2-4ac=\left(-2\right)^2-4.1.\left(-8\right)=4+32=36>0\)

phương trình có 2 nghiệm phân biệt : \(x_1=\frac{-b+\sqrt{\Delta}}{2a}=\frac{2+\sqrt{36}}{2}=\frac{2+6}{2}=\frac{8}{2}=4\)

                                                      \(x_2=\frac{-b-\sqrt{\Delta}}{2a}=\frac{2-\sqrt{36}}{2}=\frac{2-6}{2}=\frac{-4}{2}=\left(-2\right)\)

 x²+13x+42=0 

Two solutions were found :

1.  x = -6
2.  x = -7

Step by step solution :

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  x²+13x+42 

The first term is,  x²  its coefficient is  1 .
The middle term is,  +13x  its coefficient is  13 .
The last term, "the constant", is  +42 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 42 = 42 

Step-2 : Find two factors of  42  whose sum equals the coefficient of the middle term, which is   13 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  6  and  7 
                     x² + 6x + 7x + 42

Step-4 : Add up the first 2 terms, pulling out factors :
                    x • (x+6)
              Add up the last 2 terms, pulling out common factors :
                    7 • (x+6)
Step-5 : Add up the four terms of step 4 :
                    (x+7)  •  (x+6)
             Which is the desired factorization

Equation at the end of step  1  :

(x + 7) • (x + 6) = 0

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