We want to solve the given rational equation.
x/2x-1+3/x+4=21/2x^2+7x-4
We will start by factoring the denominators to find the least common denominator (LCD).
Note that the first and second denominators are already factored. Let's factor the third one.
We can now rewrite the given expression.
x/2x-1+3/x+4=21/2x^2+7x-4
⇕
x/2x-1+3/x+4=21/(x+4) (2x-1)
We will solve the equation by multiplying each side by the LCD to clear denominators.
Note that we have a quadratic equation now. Let's identify the values of a, b, and c.
x^2+10x-24=0
⇕
1x^2 +10x+(- 24)=0
We have that a= 1, b=10, and c=-24. Let's substitute these values into the Quadratic Formula and solve for x.
Let's calculate both solutions by using the positive and negative signs.
x=- 10± 14/2
x=- 10+ 14/2
x=- 10- 14/2
x=4/2
x=- 24/2
x=2
x=-12
Finally, we will check if either x=2 or x=-12 is an extraneous solution. To do this, we need to substitute 2 and - 12 for x in the original equation. Let's start with x=2.
