If $x^2+2=2^{\frac{2}{3}}+2^{-\frac{2}{3}}$, show that $2 x\left(x^2+3\right)=3$
Answer (1)
2 x ( x 2 + 3 ) = 3 is true for x = 2 3 1 − 2 − 3 1 .
Explanation
Problem Setup We are given that x 2 + 2 = 2 3 2 + 2 − 3 2 , and we want to show that 2 x ( x 2 + 3 ) = 3 .
Substitution Let's denote u = 2 3 1 . Then, the given equation can be written as x 2 + 2 = u 2 + u 2 1 .
Rewriting the Equation We can rewrite the equation as x 2 = u 2 + u 2 1 − 2 . Notice that the right side can be factored as a perfect square: x 2 = ( u − u 1 ) 2 .
Solving for x Taking the square root of both sides, we get x = ± ( u − u 1 ) = ± ( 2 3 1 − 2 − 3 1 ) .
Positive Case Now, let's consider the positive case: x = 2 3 1 − 2 − 3 1 . We want to show that 2 x ( x 2 + 3 ) = 3 . Substituting the value of x , we have:
2 x ( x 2 + 3 ) = 2 ( 2 3 1 − 2 − 3 1 ) (( 2 3 1 − 2 − 3 1 ) 2 + 3 )
Expanding the Square Expanding the expression inside the parenthesis:
( 2 3 1 − 2 − 3 1 ) 2 = ( 2 3 2 − 2 ( 2 3 1 ) ( 2 − 3 1 ) + 2 − 3 2 ) = 2 3 2 − 2 + 2 − 3 2
So, ( 2 3 1 − 2 − 3 1 ) 2 + 3 = 2 3 2 − 2 + 2 − 3 2 + 3 = 2 3 2 + 1 + 2 − 3 2
Simplifying the Expression Now, we have:
2 x ( x 2 + 3 ) = 2 ( 2 3 1 − 2 − 3 1 ) ( 2 3 2 + 1 + 2 − 3 2 ) = 2 ( 2 3 1 × 2 3 2 + 2 3 1 + 2 3 1 × 2 − 3 2 − 2 − 3 1 × 2 3 2 − 2 − 3 1 − 2 − 3 1 × 2 − 3 2 ) = 2 ( 2 + 2 3 1 + 2 − 3 1 − 2 3 1 − 2 − 3 1 − 2 − 1 ) = 2 ( 2 − 2 − 1 ) = 2 ( 2 − 2 1 ) = 2 ( 2 3 ) = 3
Result for Positive Case Thus, for x = 2 3 1 − 2 − 3 1 , we have 2 x ( x 2 + 3 ) = 3 .
Negative Case Now, let's consider the negative case: x = − ( 2 3 1 − 2 − 3 1 ) . Then, 2 x ( x 2 + 3 ) = 2 ( − ( 2 3 1 − 2 − 3 1 )) (( − ( 2 3 1 − 2 − 3 1 ) ) 2 + 3 ) = − 2 ( 2 3 1 − 2 − 3 1 ) (( 2 3 1 − 2 − 3 1 ) 2 + 3 ) .
Since we already know that ( 2 3 1 − 2 − 3 1 ) 2 + 3 = 2 3 2 + 1 + 2 − 3 2 , we have:
2 x ( x 2 + 3 ) = − 2 ( 2 3 1 − 2 − 3 1 ) ( 2 3 2 + 1 + 2 − 3 2 ) = − 3
Choosing the Correct Case However, the problem asks us to show that 2 x ( x 2 + 3 ) = 3 . Therefore, we must choose the positive case, x = 2 3 1 − 2 − 3 1 .
Final Answer Therefore, 2 x ( x 2 + 3 ) = 3 is only true for x = 2 3 1 − 2 − 3 1 .
Examples
This problem demonstrates how algebraic manipulation and substitution can simplify complex expressions. In real-world scenarios, similar techniques are used in engineering to simplify circuit analysis or in physics to solve equations of motion. For instance, when analyzing an electrical circuit, you might encounter complex impedances. By using appropriate substitutions and algebraic simplifications, you can reduce the complexity and solve for the currents and voltages in the circuit. Similarly, in physics, simplifying equations of motion can help in understanding the trajectory of projectiles or the behavior of oscillating systems. The ability to manipulate and simplify equations is a fundamental skill in many scientific and engineering fields.
