(x+3)/(x^2-x-12) + (2/x + 3)= (1)/(x-4)
(X+3)/(X^2-X-12) + 2/(X+3) = 1/(X-4).
(X+3)/(X-4)(X+3) + 2/(X+3) = 1/(X-4)
Multlply both sides by (X-4)(X+3) and get:
X+3+2(X-4) = X+3,
X+3 + 2X-8 = X+3,
X+2X-X = 3-3+8,
2X = 8,
X = 4.
Julie, there's something wrong with your Eq; because when I plug 4 into
your Eq,the 1st and last terms are
undefined. Maxe sure there is no error in your textbook.
To solve this equation, we need to find the values of x that make the equation true. Let's go through the steps:
1. Simplify the equation: To simplify, we can combine the fractions on the left side of the equation by finding a common denominator:
(x+3)/(x^2-x-12) + (2/x) + 3 = 1/(x-4)
The common denominator is (x^2-x-12)(x-4). We multiply the first fraction by (x-4) and the second fraction by (x^2-x-12):
(x+3)(x-4) + 2(x^2-x-12) + 3(x^2-x-12) = 1
Expanding and simplifying the equation gives us:
(x^2-4x+3x-12) + (2x^2-2x-24) + (3x^2-3x-36) = 1
Combine like terms gives us:
6x^2 - 6x - 85 = 1
2. Rearrange the equation: We need to move all terms to one side of the equation to simplify further:
6x^2 - 6x - 85 - 1 = 0
Simplify:
6x^2 - 6x - 86 = 0
3. Solve the quadratic equation: To solve a quadratic equation, we can factor it or use the quadratic formula. In this case, factoring may not be possible or convenient, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 6, b = -6, and c = -86. Plug these values into the quadratic formula:
x = (-(-6) ± √((-6)^2 - 4(6)(-86))) / (2(6))
Simplify:
x = (6 ± √(36 + 2064)) / 12
x = (6 ± √2100) / 12
x = (6 ± √(100 * 21)) / 12
x = (6 ± 10√21) / 12
4. Simplify the solution: We can simplify the expression further by factoring out a common factor:
x = (6 ± 10√21) / 12
x = (3 ± 5√21) / 6
These are the two possible solutions for x.
Therefore, the solutions for the given equation are x = (3 + 5√21) / 6 and x = (3 - 5√21) / 6.