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.
Let's start by simplifying the equation. To do that, we need to multiply both sides of the equation by the common denominator of the two rational expressions, which is (x - 3) * x.
(x + 5)/(x - 3) = (2x + 7)/(x)
Now, multiply both sides of the equation by the common denominator:
(x + 5)(x) = (2x + 7)(x - 3)
Expand both sides:
x^2 + 5x = 2x^2 - 6x + 7x - 21
Combine like terms:
x^2 + 5x = 2x^2 + x - 21
Move all terms to one side by subtracting the expression on the right from the expression on the left:
0 = 2x^2 + x - 21 - x^2 - 5x
Simplify:
0 = x^2 - 4x - 21
Now, we have a quadratic equation. To solve it, we can either factorize or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In our case, a = 1, b = -4, and c = -21:
x = (-(-4) ± sqrt((-4)^2 - 4(1)(-21))) / (2(1))
Simplify:
x = (4 ± sqrt(16 + 84)) / 2
x = (4 ± sqrt(100)) / 2
x = (4 ± 10) / 2
Now we have two possible solutions:
x = (4 + 10) / 2 = 14 / 2 = 7
x = (4 - 10) / 2 = -6 / 2 = -3
Therefore, the solutions to the rational function equation are x = 7 and x = -3.