1)
[tex]\dfrac{x^2-16}{x^2+6x+8}[/tex]
Decompose the numerator and denominator into multipliers
To simplify the numerator we use the formula of difference of squares
[tex]x^2-y^2=(x-y)(x+y)[/tex]
[tex]x^2-16=(x-4)(x+4)[/tex]
To decompose the denominator into multipliers solve the square equation
[tex]x^2+6x+8=0\\D=6^2-4*8=4=2^2\\x_1=\dfrac{-6+2}{2} =-2\\x_2=\dfrac{-6-2}{2} =-4[/tex]
Formula for factoring a square equation
[tex](x-x_1)(x-x_2)[/tex]
Substituting the found roots of the equation into the formula
[tex](x-(-2))(x-(-4))=(x+2)(x+4)[/tex]
After simplifying the numerator and denominator we get a fraction
[tex]\dfrac{(x-4)(x+4)}{(x+2)(x+4)}=\dfrac{x-4}{x+2}[/tex], so
[tex]\dfrac{x^2-16}{x^2+6x+8}=\dfrac{(x-4)(x+4)}{(x+2)(x+4)}=\dfrac{x-4}{x+2}[/tex]
2)
[tex]\dfrac{x^2-x-6}{x^2-3x-10}[/tex]
Decompose the numerator and denominator into multipliers
To decompose the numerator into multipliers solve the square equation
[tex]x^2-x-6=0\\D=(-1)^2-4*(-6)=25=5^2\\x_1=\dfrac{1+5}{2} =3\\x_2=\dfrac{1-5}{2} =-2[/tex]
Formula for factoring a square equation
[tex](x-x_1)(x-x_2)[/tex]
Substituting the found roots of the equation into the formula
[tex](x-3)(x-(-2))=(x-3)(x+2)[/tex]
To decompose the denominator into multipliers solve the square equation
[tex]x^2-3x-10=0\\D=(-3)^2-4*(-10)=49=7^2\\x_1=\dfrac{3+7}{2} =5\\x_2=\dfrac{3-7}{2} =-2[/tex]
Formula for factoring a square equation
[tex](x-x_1)(x-x_2)[/tex]
Substituting the found roots of the equation into the formula
[tex](x-5)(x-(-(-2))=(x-5)(x+2)[/tex]
After simplifying the numerator and denominator we get a fraction
[tex]\dfrac{(x-3)(x+2)}{(x-5)(x+2)}=\dfrac{x-3}{x-5}[/tex], so
[tex]\dfrac{x^2-x-6}{x^2-3x-10}=\dfrac{(x-3)(x+2)}{(x-5)(x+2)}=\dfrac{x-3}{x-5}[/tex]
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