Simplifying rational expressions x^2-16/x^2+6x+8 = x-4/x+2 x^2-x-6/x^2-3x-10 = x-3/x-5

Question

asked Aug 3, 2021 39.3k views

2 Answers

Fza · Answer 1 · 2021-08-09T19:22:50+0000

1)

(x^2-16)/(x^2+6x+8)

Decompose the numerator and denominator into multipliers

To simplify the numerator we use the formula of difference of squares

x^2-y^2=(x-y)(x+y)

x^2-16=(x-4)(x+4)

To decompose the denominator into multipliers solve the square equation

$x^2+6x+8=0\\D=6^2-4*8=4=2^2\\x_1=(-6+2)/(2) =-2\\x_2=(-6-2)/(2) =-4$

Formula for factoring a square equation

(x-x_1)(x-x_2)

Substituting the found roots of the equation into the formula

(x-(-2))(x-(-4))=(x+2)(x+4)

After simplifying the numerator and denominator we get a fraction

((x-4)(x+4))/((x+2)(x+4))=(x-4)/(x+2) , so

(x^2-16)/(x^2+6x+8)=((x-4)(x+4))/((x+2)(x+4))=(x-4)/(x+2)

2)

(x^2-x-6)/(x^2-3x-10)

Decompose the numerator and denominator into multipliers

To decompose the numerator into multipliers solve the square equation

$x^2-x-6=0\\D=(-1)^2-4*(-6)=25=5^2\\x_1=(1+5)/(2) =3\\x_2=(1-5)/(2) =-2$

Formula for factoring a square equation

(x-x_1)(x-x_2)

Substituting the found roots of the equation into the formula

(x-3)(x-(-2))=(x-3)(x+2)

To decompose the denominator into multipliers solve the square equation

$x^2-3x-10=0\\D=(-3)^2-4*(-10)=49=7^2\\x_1=(3+7)/(2) =5\\x_2=(3-7)/(2) =-2$

Formula for factoring a square equation

(x-x_1)(x-x_2)

Substituting the found roots of the equation into the formula

(x-5)(x-(-(-2))=(x-5)(x+2)

After simplifying the numerator and denominator we get a fraction

((x-3)(x+2))/((x-5)(x+2))=(x-3)/(x-5) , so

(x^2-x-6)/(x^2-3x-10)=((x-3)(x+2))/((x-5)(x+2))=(x-3)/(x-5)

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