Question

Solve. $\sqrt[3]{x^{2}-x+18}=\sqrt[3]{2 x^{2}-3 x-6}$

   Solve.
$\sqrt[3]{x^{2}-x+18}=\sqrt[3]{2 x^{2}-3 x-6}$

Intermediate Algebra

Intermediate Algebra

Lynn Marecek 2nd Edition

Chapter 8, Problem 330

Instant Answer

Solved by Expert Alison Rodriguez

09/25/2022

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This gives us: \[x^{2}-x+18=2 x^{2}-3 x-6\] Show more…

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Solve. $\sqrt[3]{x^{2}-x+18}=\sqrt[3]{2 x^{2}-3 x-6}$

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Key Concepts

Cube Root Function

The cube root function, defined for all real numbers, gives the unique real number whose cube equals a given number. Unlike square roots, cube roots work for negative, zero, and positive inputs, and they preserve the order of numbers.

One-to-One Property

A one-to-one function is one where each input corresponds to a single unique output. Because the cube root function is one-to-one, if the cube roots of two expressions are equal, then the expressions themselves must be equal.

Radical Equations

Radical equations involve variables underneath a radical sign. With cube roots, you can safely eliminate the radical by cubing both sides of the equation due to the one-to-one nature of the cube root function, directly yielding an equation between the radicands.

Polynomial Equations

After eliminating the cube roots, the problem reduces to solving a polynomial equation. This involves manipulating and simplifying the equation, often resulting in quadratic or higher degree equations that can be solved using factoring, the quadratic formula, or other algebraic techniques.

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