Given, f(x) = 4x^2 + 6x and g(x) = 2x^2 +13x + 15 , find (f/g)(x) Given, f(x) = x^2 - 6x+ 8 and g(x) = x - 2, solve f(x) = g(x) Show Work?

Question

Given, f(x) = 4x^2 + 6x and g(x) = 2x^2 +13x + 15 , find (f/g)(x) Given, f(x) = x^2 - 6x+ 8 and g(x) = x - 2, solve f(x) = g(x) Show Work?

Answer 1

(f/g) (x) = f(x)/g(x) = (4x^2+6x) /(2x^2+13x+15)

4x^2+6x = 2x(2x+3)



2x^2+13x+15

= 2x^2+10x+3x+15

= 2x(x+5)+3(x+5)

= (x+5)(2x+3)

(f/g)(x) = 2x(2x+3) / ((x+5)(2x+3))

where x ≠ -5 and x ≠ -3/2

...... ...... ........ .............

f(x) = x^2-6x+8 = (x-4)(x-2)

g(x) = x-2

f(x) = g(x)

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

divide both sides by x-2

x-4 = 1

x= 5

Answer 2

f/g(x) = (4x^2 + 6x) / ( 2x^2 + 13x + 15)

f/g(x) = 2x(2x + 3) / (2x + 3)(x + 5)

Caancel down by '2x + 3'

f/g(x) = 2x / (x + 5)



f(x) = g(x) => x^2 - 6x + 8 = x - 2

x^2 - 7x + 10 = 0

(x - 5)(x - 2) = 0

Hence

x = 5 & x = 2

Hence

f(5) = g(5)

&

f(2) = g(2)

Answer 3

In your first question, regarding (f/g)(x), be careful about simplifying the rational function. It cannot be defined at any root of g(x). Canceling (2x + 3) from the numerator and denominator does not change that. The function still is undefined at -3/2.



(f/g)(x) = 2x/(x + 5), x ≠ -5, x ≠ -3/2



If you neglect the exclusion, you will have to settle for partial credit.

Answer 4

Work = force x distance

Answer 5

The first problem really boils down to factoring the two polynomials to see if the numerator and denominator have common factors to cancel:



4x² + 6x = 2x(2x + 3) ... This is easy

2x² + 13x + 15 = 2x² + 10x + 3x + 15 .... rewrite 13x as 10x + 3x

= 2x(x + 5) + 3(x + 5) .... factor pairs

= (2x + 3)(x + 5)



So, there's a common factor of (2x + 3) that can be eliminated.



f(x)/g(x) = 2x/(x + 5) .... after canceling



The second problem is simple quadratic equation solving,



f(x) = g(x)

x² - 6x + 8 = x - 2 .... substitute for f(x), g(x)

x² - 7x + 10 = 0 .... subtract x and add 2 to get 0 on the right



Just two steps to standard form. You should have solved a bunch like this by now.