How do you solve the rational equation #x^2/(x-3) - 5/(x-3) = 0#?

Question

Avery Alexander · Accepted Answer

#x= +-sqrt5#

#x^2/(x-3) - 5/(x-3) = 0#

First, notice that the denominator is the same for both terms, so we can combine the numerator over the common denominator.

#(x^2 - 5)/(x-3) = 0#

Now we need to contend with the #x# in the denominator. We can multiply both sides of the equation by #(x-3)#. The left hand side cancels the denominator, and the right hand side multiplies by zero.

#(x^2-5)cancel((x-3)/(x-3))^1 = 0cancel((x-3))^0#

Now we are left with;

#x^2-5 = 0#

We only have one #x# term and a constant, so lets add #5# to both sides.

#x^2 = 5#

Now we just need to get rid of the exponent. We can take the square root of both sides to get our #x# values;

#x= +-sqrt5#

Kennedy Adams · Answer

undefined To solve the rational equation x^2/(x-3) - 5/(x-3) = 0, we can combine the fractions on the left side by finding a common denominator. In this case, the common denominator is (x-3).

After combining the fractions, we have (x^2 - 5)/(x-3) = 0.

To solve for x, we can set the numerator equal to zero: x^2 - 5 = 0.

Next, we can factor the quadratic equation: (x - √5)(x + √5) = 0.

Setting each factor equal to zero, we have two possible solutions: x - √5 = 0 or x + √5 = 0.

Solving for x in each equation, we get x = √5 or x = -√5.

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