Final answer:
To find the value(s) of x for the given rational equation, we must find a common denominator, create a quadratic equation, and solve it using the quadratic formula, while discarding any non-sensical or undefined values.
Step-by-step explanation:
The task is to find the value(s) of x that satisfy the rational equation (x)/(x+2)+(x)/(3)=(-6)/(3x+6). This problem can be approached by finding a common denominator and combining terms to form a quadratic equation that can be solved for x. However, we must make sure to discard any solution that makes no physical sense (such as a negative concentration in a chemistry context), or any values that make the denominators zero (since division by zero is undefined).
To solve this equation, first find a common denominator for the three terms, which is (x+2)×3(x+6). After combining the terms and simplifying the equation, it will take the form of a quadratic equation ax² + bx + c = 0, where a, b, and c are constants. This quadratic equation can be solved using the quadratic formula x = (-b ± √(b² - 4ac))/(2a) to find the two possible values of x.
When applying the quadratic formula, we will obtain two solutions, and we need to evaluate both signs in the numerator—first the + sign and then the - sign. After solving, we must check for any extraneous solutions and discard them. For example, if a problem involves concentrations, we cannot have a negative concentration, so any negative solution must be excluded.