- Identifying the Quadratic Equation
Given quadratic equation: \( \frac{3}{x^2} + \frac{14}{x} + 8 = 0 \)
- Splitting the Middle Term
To split the middle term, we need to find two numbers that multiply to the product of the leading coefficient (3) and the constant term (8), which is 24, and add up to the middle coefficient (14).
The two numbers that satisfy these conditions are 6 and 4 because 6 * 4 = 24 and 6 + 4 = 10.
- Restructuring the Equation
We can rewrite the middle term of the equation using the numbers 6 and 4:
\( \frac{3}{x^2} + \frac{6}{x} + \frac{8}{x} + 8 = 0 \)
Now, we can group the terms:
\( (\frac{3}{x^2} + \frac{6}{x}) + (\frac{8}{x} + 8) = 0 \)
- Factoring by Grouping
We can factor out common terms from each group:
\( \frac{3}{x}(x + 2) + \frac{8}{x}(x + 1) = 0 \)
- Final Equation
After factoring by grouping, we get:
\( \frac{3(x + 2)}{x} + \frac{8(x + 1)}{x} = 0 \)
Now, we can solve for x by setting each factor to zero:
\( x + 2 = 0 \) or \( x + 1 = 0 \)
Therefore, the solutions are \( x = -2 \) and \( x = -1 \).