Question

Part A: Solve the rational equation 1 / x−4+ x// x−2 = 2 / x²−6x+8

Part B: Solve the radical equation x+11−x=−1
Part C: Compare and contrast the processes of solving rational and radical equations. Use the equations in parts A and B to illustrate your answer, if desired.
a) x=2,5; x=−3; Differences in complexity and solutions.
b) x=4,2; x=−12; Similarities in approach, differences in solutions.
c) x=2,3; x=−11; No notable differences.
d) x=3,4; x=−5; Differences in approach and solutions.

Answer 1

Part A: The rational equation has no real solutions. Part B: The radical equation has no solution. Part C: Rational and radical equations have different solving processes and may have different solutions depending on the equation.

Step-by-step explanation:

Part A:

To solve the rational equation, we need to combine the fractions on the left side with a common denominator. In this case, the common denominator is x(x-2). The equation becomes:

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

Multiplying each term by x(x-2), we get:

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

Expanding and simplifying the equation gives us a quadratic equation:

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

Simplifying further, we get:

2x^2 - 4x = 2x - 8

2x^2 - 6x + 8 = 0

Using the quadratic formula, we can solve for x:

x = (6±√(6^2-4(2)(8)))/(2(2))

x = (6±√(36-64))/4

x = (6±√(-28))/4

Since the discriminant is negative, there are no real solutions for this equation.

Part B:

To solve the radical equation, we need to isolate the radical term on one side of the equation. In this case, we have:

x + 11 - x = -1

Combining like terms, we get:

11 = -1

This is a contradiction and has no solution.

Part C:

The process of solving rational equations involves finding a common denominator and simplifying the equation to a quadratic form. On the other hand, solving radical equations involves isolating the radical term and simplifying the equation. While both types of equations may involve solving quadratic equations, the techniques used are different. In the given examples, the rational equation had no real solutions, while the radical equation had a contradiction and also had no real solutions.