Question
Solve the equation, eliminating any extraneous solutions.
x^2+5/(x+5) = 30/(x+5)
x^2+5/(x+5) = 30/(x+5)
Answers
GPT 3.5
To solve the equation, we want to first get rid of the denominators by multiplying both sides of the equation by (x+5):
(x^2 + 5)/(x+5) = 30/(x+5)
(x^2 + 5) = 30
Now, subtract 30 from both sides:
x^2 + 5 - 30 = 0
x^2 - 25 = 0
Now, we can factor the left side of the equation:
(x+5)(x-5) = 0
Setting each factor to zero:
x+5=0 or x-5=0
x=-5 x=5
So, the solution to the equation is x = -5 or x = 5.
To eliminate any extraneous solutions, we have to check if either solution makes the original equation undefined. Since the original equation had a denominator of (x+5), x = -5 will make the denominator zero. Therefore, x = -5 is an extraneous solution. So, the only valid solution to the equation is x = 5.
(x^2 + 5)/(x+5) = 30/(x+5)
(x^2 + 5) = 30
Now, subtract 30 from both sides:
x^2 + 5 - 30 = 0
x^2 - 25 = 0
Now, we can factor the left side of the equation:
(x+5)(x-5) = 0
Setting each factor to zero:
x+5=0 or x-5=0
x=-5 x=5
So, the solution to the equation is x = -5 or x = 5.
To eliminate any extraneous solutions, we have to check if either solution makes the original equation undefined. Since the original equation had a denominator of (x+5), x = -5 will make the denominator zero. Therefore, x = -5 is an extraneous solution. So, the only valid solution to the equation is x = 5.