Using rational functions solve
7+ (1)/x = 1/(x-2)
To solve the given rational function equation, 7 + 1/x = 1/(x - 2), we need to find the value(s) of x that make the equation true.
Here's how you can solve it step by step:
Step 1: Get rid of the denominators by finding a common denominator.
In this case, the common denominator is x(x - 2). Multiply every term on both sides of the equation by x(x - 2):
x(x - 2) * (7 + 1/x) = x(x - 2) * (1/(x - 2))
Simplifying this gives us:
7x(x - 2) + (x - 2) = x
Step 2: Expand and simplify the equation.
Expand the terms on the left side of the equation:
7x^2 - 14x + x - 2 = x
Combine like terms:
7x^2 - 13x - 2 = x
Step 3: Move all terms to one side of the equation.
Subtract x from both sides:
7x^2 - 14x - 2 = 0
Step 4: Solve the quadratic equation.
This equation is a quadratic equation, so we can solve it by factoring, completing the square, or using the quadratic formula. However, the given equation does not factor easily, so let's use the quadratic formula:
The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 7, b = -14, and c = -2. Substituting these values into the quadratic formula, we get:
x = (-(-14) ± √((-14)^2 - 4 * 7 * (-2))) / (2 * 7)
Simplifying further, we have:
x = (14 ± √(196 + 56)) / 14
x = (14 ± √252) / 14
Step 5: Simplify the square root and solve for x.
To simplify the square root, we can break down 252 into its prime factors. The square root of 252 simplifies to 2√63.
Therefore, we have:
x = (14 ± 2√63) / 14
Simplifying the fraction:
x = (7 ± √63) / 7
So, the two solutions to the rational function equation 7 + 1/x = 1/(x - 2) are:
x = (7 + √63) / 7
x = (7 - √63) / 7
Remember that division by zero is undefined, so be sure to check if either of the solutions makes the denominator zero. In this case, (x - 2) should not equal zero. Therefore, the solution x = (7 + √63) / 7 is valid, while x = (7 - √63) / 7 is not valid in this context.