Final answer:
The least common denominator of the equation 1/x + 2/(x+10) = 1/3 is x(x+10). After multiplying each term by the LCD to eliminate the denominators and simplifying, we find a quadratic equation; its discriminant is positive, which means there are two valid solutions.
Step-by-step explanation:
To solve the equation 1/x + 2/(x+10) = 1/3, we first need to find the least common denominator (LCD) of the fractions. Since x and x+10 are different and cannot be factored into the same terms, the LCD will simply be the product of these two expressions, so the least common denominator is x(x+10).
Next, we'll express each fraction with the LCD as the denominator:
We then multiply both sides of the equation by 3x(x+10) to eliminate the denominators:
- 3(3x+10) = x(x+10)
- 9x + 30 = x2 + 10x
- x2 + 10x - 9x - 30 = 0
- x2 + x - 30 = 0
Solving this quadratic equation will give us the valid solutions for x. If the discriminant (b2 - 4ac) is positive, there will be two valid solutions; if it is zero, one valid solution; and if it's negative, no real solutions. After calculating the discriminant, we find that it's positive, thus the equation will have two valid solutions.
Finally, eliminate terms wherever possible to simplify the algebra. After simplifications, we check the solutions to ensure they are reasonable and do not result in division by zero.
Complete Question:
Select the correct answer from each drop-down menu.
Consider this equation. 1/x + 2/x+10 = 1/3
Complete the statements to make them true.
The least common denominator is _
The equation will have _ valid solutions.