Using rational functions solve
x+5(numerator) x-3 (denominator) = 2x+7 (numerator) x(demominator)
To solve the rational function equation, we need to find the values of x that make the equation true. Here's how to do it:
1. Start by multiplying both sides of the equation by the common denominator, which in this case is (x - 3) * x. This will help eliminate the denominators in the equation.
(x + 5) * x = (2x + 7) * (x - 3)
Simplify the equation:
x^2 + 5x = 2x^2 - 6x + 7x - 21
Combine like terms:
x^2 + 5x = 2x^2 + x - 21
2. Bring all the terms to one side of the equation to set it equal to zero:
x^2 + 5x - 2x^2 - x + 21 = 0
Rearrange the terms:
-x^2 + 4x + 21 = 0
3. Now we have a quadratic equation in standard form: ax^2 + bx + c = 0. We can solve it by factoring, completing the square, or using the quadratic formula.
Since the coefficient of the x^2 term is negative (-1), it cannot be easily factored using integer values. So, let's use the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a).
Here, a = -1, b = 4, and c = 21. Substituting these values into the formula:
x = (-4 ± √(4^2 - 4(-1)(21)))/(2(-1))
Simplify:
x = (-4 ± √(16 + 84)) / (-2)
x = (-4 ± √100) / (-2)
x = (-4 ± 10) / (-2)
4. Compute both possible values of x:
For x = (-4 + 10) / (-2):
x = 6 / (-2)
x = -3
For x = (-4 - 10) / (-2):
x = -14 / (-2)
x = 7
5. Therefore, the solution to the rational function equation is x = -3 or x = 7.