Which is the correct solution to the cubic equation, $\sqrt[3]{2 x-3}=-3$?
A. 12
B. 6
C. -12
D. -6
Answer (2)
Cube both sides of the equation: ( 3 2 x − 3 ) 3 = ( − 3 ) 3 .
Simplify the equation: 2 x − 3 = − 27 .
Isolate x : 2 x = − 24 .
Solve for x : x = − 12 . The solution is − 12 .
Explanation
Understanding the Problem We are given the equation 3 2 x − 3 = − 3 . Our goal is to find the value of x that satisfies this equation.
Cubing Both Sides To eliminate the cube root, we cube both sides of the equation: ( 3 2 x − 3 ) 3 = ( − 3 ) 3
Simplifying the Equation Simplifying both sides, we get: 2 x − 3 = − 27
Adding 3 to Both Sides Next, we add 3 to both sides of the equation to isolate the term with x : 2 x = − 27 + 3
Simplifying Simplifying further: 2 x = − 24
Dividing by 2 Now, we divide both sides by 2 to solve for x : x = 2 − 24
Calculating the Value of x Finally, we calculate the value of x : x = − 12 Therefore, the correct solution to the cubic equation is − 12 .
Examples
Cubic equations can be used to model various real-world phenomena, such as the volume of a container or the trajectory of a projectile. For example, if you were designing a rectangular container with specific volume requirements and the dimensions were related by a cubic equation, solving the equation would help you determine the exact measurements needed. Similarly, in physics, cubic equations can arise when analyzing motion under certain conditions, allowing you to predict the position of an object at a given time.
The solution to the cubic equation 3 2 x − 3 = − 3 is x = − 12 . Therefore, the correct answer is C . − 12 .
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