Fraction equations

Question

(X/x + 5) - (3x/x + 2) = (4/x^2 + 7x + 10)

solve for x

Answer 1

First thing you might notice is that the denominator on the right side is the product of the two denominators on the left side.

That's assuming you got the parentheses slightly in the wrong place, but it's too much of a coincidence.

Put the left side over the same denominator (x+2)(x+7). Those are the factors of the denominator on the right side.

Now add the two fractions on the left side, once you get the common denominator

that leads to the left side numerator as x(x+2) - 3x(x+5) = x^2 +2x -3x^2 -15x = -2x^2-13x

set that equal to the numerator on the right side: 4

-2x^2 -13x = 4

move the 4 to the left side and use the quadratic equation to solve for x

-2x^2 -13x - 4 = 0

multiply through by -1

2x^2 +13x + 4 = 0

x = -13/4 + or - (1/4)square root of 169-4(2)(4)

= --13/4+ or - (1/4)sqr(137)

Answer 2

x - 3x = 4

x + 5 x + 2 x² + 7x + 10

Your first step is to find a common denominator, which turns out to be (x + 5)(x + 2) [if you factor x² + 7x + 10, you get (x + 2)(x + 5)]

To get all terms to have the same denominator, multiply the first term by (x + 2)/(x + 2) and the second term by (x + 5)/(x + 5).

x • (x + 2) - 3x • (x + 5) = 4

x + 5 (x + 2) x + 2 (x + 5) x² + 7x + 10

x² + 2x - 3x² + 15x = 4

(X + 5)(x + 2) (X + 5)(x + 2) (x + 5)(x + 2)

Since all the denominators are the same, you can just work with the numerator:

x² + 2x - (3x² + 15x) = 4

x² + 2x - 3x² - 15x = 4

-2x² - 13x = 4

Bring everything to one side of the equation

2x² + 13x + 4 = 0

Solve using quadratic formula

-13 +/- √(13² - 4(2)(4)) = -13 +/- √(169 - 32) = -13 +/- √(137)

2(2) 4 4

x = -13 + √(137) , -13 - √(137)

4 4

Hope this helps. It's hard to write without proper formatting.