\frac{x+4}{x-2}-\frac{x-2}{x}+4=0
Answer (2)
To solve the equation x − 2 x + 4 − x x − 2 + 4 = 0 , we eliminate fractions by multiplying through by the least common denominator, simplify it to a quadratic equation 4 x 2 − 4 = 0 , and find the solutions are x = 1 and x = − 1 . Both solutions are verified as valid by substituting them back into the original equation. Therefore, the complete solution is x = 1 and x = − 1 .
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Multiply both sides by x ( x − 2 ) to eliminate fractions.
Expand and simplify the equation to 4 x 2 − 4 = 0 .
Solve the quadratic equation x 2 − 1 = 0 by factoring, obtaining x = 1 and x = − 1 .
Verify that both solutions are valid by substituting them back into the original equation, thus the solutions are x = 1 , x = − 1 .
Explanation
Problem Analysis We are given the equation x − 2 x + 4 − x x − 2 + 4 = 0 and we need to find the value(s) of x that satisfy this equation.
Eliminating Fractions To solve the equation, we first eliminate the fractions by multiplying both sides of the equation by x ( x − 2 ) . This gives us: x ( x − 2 ) ( x − 2 x + 4 − x x − 2 + 4 ) = 0 x ( x + 4 ) − ( x − 2 ) ( x − 2 ) + 4 x ( x − 2 ) = 0
Expanding the Terms Expanding the terms, we get: x 2 + 4 x − ( x 2 − 4 x + 4 ) + 4 x 2 − 8 x = 0 x 2 + 4 x − x 2 + 4 x − 4 + 4 x 2 − 8 x = 0
Combining Like Terms Combining like terms, we have: 4 x 2 + ( x 2 − x 2 ) + ( 4 x + 4 x − 8 x ) − 4 = 0 4 x 2 − 4 = 0
Simplifying the Equation Dividing both sides by 4, we get: x 2 − 1 = 0
Factoring the Quadratic Factoring the quadratic, we have: ( x − 1 ) ( x + 1 ) = 0
Finding the Solutions Thus, the solutions are x = 1 and x = − 1 . We need to check if these solutions are extraneous by substituting them back into the original equation.
Checking for Extraneous Solutions For x = 1 :
1 − 2 1 + 4 − 1 1 − 2 + 4 = − 1 5 − 1 − 1 + 4 = − 5 + 1 + 4 = 0 So, x = 1 is a valid solution. For x = − 1 :
− 1 − 2 − 1 + 4 − − 1 − 1 − 2 + 4 = − 3 3 − − 1 − 3 + 4 = − 1 − 3 + 4 = 0 So, x = − 1 is also a valid solution.
Final Answer Therefore, the solutions to the equation are x = 1 and x = − 1 .
Examples
When designing structures, engineers often encounter equations similar to the one we solved. For example, when analyzing the stability of a bridge or the forces acting on a building, they might need to solve rational equations to ensure the structure can withstand various loads. The ability to manipulate and solve these equations is crucial for ensuring safety and efficiency in engineering designs. The solutions x = 1 and x = − 1 could represent critical points or parameters in such a design, influencing decisions about material selection or structural dimensions.
