# How do you solve #x^2 / (x+3) - 5 / (x+3) = 0# and find any extraneous solutions?

We can cross-multiply since there are only two fractions, so let's move one to the other side.

But we could have found the solution much more quickly if we had taken the fraction into consideration from the start!

Since the denominators and numerators are equal, they must also be.

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To solve the equation x^2 / (x+3) - 5 / (x+3) = 0 and find any extraneous solutions, we can start by combining the fractions on the left side of the equation with a common denominator of (x+3). This gives us (x^2 - 5) / (x+3) = 0.

Next, we can set the numerator equal to zero and solve for x. So, x^2 - 5 = 0.

Using the difference of squares formula, we can factor this equation as (x - √5)(x + √5) = 0.

Setting each factor equal to zero, we have x - √5 = 0 and x + √5 = 0.

Solving for x in each equation, we find x = √5 and x = -√5.

Now, we need to check if these solutions are extraneous by substituting them back into the original equation.

When we substitute √5 into the equation, we get (√5)^2 / (√5 + 3) - 5 / (√5 + 3) = 0. Simplifying this, we get 5 / (√5 + 3) - 5 / (√5 + 3) = 0.

Since the denominators are the same, the fractions cancel out, resulting in 0 = 0.

Similarly, when we substitute -√5 into the equation, we get (-√5)^2 / (-√5 + 3) - 5 / (-√5 + 3) = 0. Simplifying this, we get 5 / (-√5 + 3) - 5 / (-√5 + 3) = 0.

Again, the fractions cancel out, resulting in 0 = 0.

Since both solutions satisfy the original equation, there are no extraneous solutions.

Therefore, the solutions to the equation x^2 / (x+3) - 5 / (x+3) = 0 are x = √5 and x = -√5.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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