In two or more complete sentences, explain how to solve the cube root equation, x-1+3=0
Answer (2)
Isolate the cube root term: 3 x − 1 = − 3 .
Cube both sides: ( 3 x − 1 ) 3 = ( − 3 ) 3 , which simplifies to x − 1 = − 27 .
Solve for x : x = − 27 + 1 = − 26 .
The solution is − 26 .
Explanation
Understanding the Problem We are given the cube root equation 3 x − 1 + 3 = 0 . Our goal is to isolate x and find its value.
Isolating the Cube Root First, we need to isolate the cube root term. To do this, subtract 3 from both sides of the equation: 3 x − 1 + 3 − 3 = 0 − 3 3 x − 1 = − 3
Cubing Both Sides Now that we have the cube root isolated, we can eliminate it by cubing both sides of the equation: ( 3 x − 1 ) 3 = ( − 3 ) 3
Simplifying the Equation Simplify both sides of the equation. Remember that ( − 3 ) 3 = − 3 × − 3 × − 3 = − 27 . So we have: x − 1 = − 27
Isolating x Next, isolate x by adding 1 to both sides of the equation: x − 1 + 1 = − 27 + 1 x = − 26
Checking the Solution Finally, we should check our solution by substituting x = − 26 back into the original equation: 3 − 26 − 1 + 3 = 3 − 27 + 3 Since 3 − 27 = − 3 , we have: − 3 + 3 = 0 The solution satisfies the original equation. Therefore, our solution is correct.
Final Answer The solution to the cube root equation 3 x − 1 + 3 = 0 is x = − 26 .
Examples
Cube root equations appear in various fields, such as physics and engineering, when dealing with volumes and scaling. For instance, if you're designing a cubic container with a specific volume, you might need to solve a cube root equation to determine the side length. Similarly, in image processing, cube root scaling can be used to adjust the intensity of colors. Understanding how to solve these equations allows for precise calculations and adjustments in these real-world applications.
To solve the equation 3 x − 1 + 3 = 0 , first isolate the cube root to get 3 x − 1 = − 3 , then cube both sides to find x − 1 = − 27 , leading to x = − 26 . After verification, this solution satisfies the original equation. Thus, the final answer is x = − 26 .
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