(1)/(x-6)+(x)/(x-2)=(4)/(x^(2)-8x+12)

Jasonbradley

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Jun 16, 2024

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Solving the Rational Equation: (1)/(x-6)+(x)/(x-2)=(4)/(x^(2)-8x+12)

This article will guide you through the steps of solving the rational equation:

(1)/(x-6)+(x)/(x-2)=(4)/(x^(2)-8x+12)

1. Factor the denominator of the right side

The denominator on the right side of the equation can be factored as a difference of squares:

(4)/(x^(2)-8x+12) = (4)/((x-6)(x-2))

2. Find a common denominator

To add the fractions on the left side, we need to find a common denominator. The least common multiple of (x-6) and (x-2) is (x-6)(x-2).

We can rewrite the equation as:

(1)/(x-6) * ((x-2)/(x-2)) + (x)/(x-2) * ((x-6)/(x-6)) = (4)/((x-6)(x-2))

3. Simplify the equation

Now, we can simplify the equation by multiplying the numerators and denominators:

(x-2)/( (x-6)(x-2) ) + (x(x-6))/( (x-2)(x-6) ) = (4)/((x-6)(x-2))

(x-2 + x(x-6))/( (x-6)(x-2) ) = (4)/((x-6)(x-2))

4. Solve for x

Since the denominators are the same, we can equate the numerators:

(x-2 + x(x-6)) = 4

x - 2 + x^2 - 6x = 4

x^2 - 5x - 6 = 4

x^2 - 5x - 10 = 0

Now, we have a quadratic equation. We can solve for x using the quadratic formula:

x = (5 ± √(5^2 - 4 * 1 * -10)) / (2 * 1)

x = (5 ± √65) / 2

Therefore, the solutions to the equation are:

x = (5 + √65) / 2 and x = (5 - √65) / 2

5. Check for extraneous solutions

It's important to check if these solutions are valid by plugging them back into the original equation. We need to ensure that the denominator does not become zero.

In this case, both solutions are valid and do not make the denominator zero.

Conclusion

The solution to the rational equation (1)/(x-6)+(x)/(x-2)=(4)/(x^(2)-8x+12) are x = (5 + √65) / 2 and x = (5 - √65) / 2.