Solve: [tex]$\sqrt[3]{x^2-8}=2$[/tex]
Answer (2)
The solutions to the equation 3 x 2 − 8 = 2 are x = 4 and x = − 4 . Both values satisfy the original equation when verified. Thus, we conclude that both solutions are valid.
;
Cube both sides of the equation to get rid of the cube root: x 2 − 8 = 8 .
Add 8 to both sides: x 2 = 16 .
Take the square root of both sides: x = ± 4 .
Verify both solutions in the original equation, confirming that x = − 4 and x = 4 are valid, so the final answer is x = − 4 or x = 4 .
Explanation
Understanding the Problem We are given the equation 3 x 2 − 8 = 2 and asked to solve for x . This involves isolating x by first eliminating the cube root.
Eliminating the Cube Root To eliminate the cube root, we cube both sides of the equation: ( 3 x 2 − 8 ) 3 = 2 3 This simplifies to: x 2 − 8 = 8
Isolating x 2 Next, we isolate x 2 by adding 8 to both sides of the equation: x 2 − 8 + 8 = 8 + 8 x 2 = 16
Solving for x Now, we take the square root of both sides of the equation to solve for x : x = ± 16 This gives us two possible solutions: x = 4 or x = − 4
Verifying the Solutions We need to verify these solutions by substituting them back into the original equation.
For x = 4 : 3 ( 4 ) 2 − 8 = 3 16 − 8 = 3 8 = 2 This solution is valid.
For x = − 4 : 3 ( − 4 ) 2 − 8 = 3 16 − 8 = 3 8 = 2 This solution is also valid.
Final Answer Therefore, the solutions to the equation 3 x 2 − 8 = 2 are x = 4 and x = − 4 .
Examples
Radical equations are used in various fields such as physics and engineering to model relationships between variables. For example, the period of a pendulum can be modeled using a radical equation, where the period depends on the length of the pendulum. Solving such equations helps in determining the required length for a specific period or vice versa. Similarly, in electrical engineering, radical equations can appear when analyzing circuits involving inductors and capacitors, where the impedance depends on the square root of certain parameters.
