Question

Solve each equation, if possible. $$ w\left(4-w^{2}\right)=8-w^{3} $$

Short answer

The solution is \(w = 2\).

Step-by-step solution

Rewrite the equation

Start by rewriting the given equation: \(w(4 - w^2) = 8 - w^3\). Distribute \(w\) on the left side to obtain: \(4w - w^3\). Now our equation is: \(4w - w^3 = 8 - w^3\).

Isolate terms involving \(w\)

Next, add \(w^3\) to both sides to eliminate \(-w^3\) from the equation: \[4w - w^3 + w^3 = 8 - w^3 + w^3\], which simplifies to \(4w = 8\).

Solve for \(w\)

To solve for \(w\), divide both sides by 4: \(w = \frac{8}{4}\), which simplifies to \(w = 2\).

Verify the solution

Check the solution by substituting \(w=2\) back into the original equation: \(2(4 - 2^2) = 8 - 2^3\). This simplifies to \(2(4 - 4) = 8 - 8\) or \(2(0) = 0\), which is true. Therefore, \(w = 2\) is a valid solution.

Key concepts

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols.
In algebra, letters like \(w\) represent numbers. This allows us to work with equations and formulas to find unknown values.
In the given problem, we use algebraic techniques to transform and solve the given equation.
Algebra helps us to simplify expressions, isolate variables, and solve equations.
For example, in the step-by-step solution, we rearranged the equation \(4w - w^3 = 8 - w^3\) to make it easier to solve by isolating \(w\).

Polynomial Equations

A polynomial equation is an equation that involves terms with variables raised to whole-number exponents and coefficients.
They can look like \(ax^n + bx^{n-1} + ... + c = 0\).
The given equation \(w(4 - w^2) = 8 - w^3\) is a polynomial equation once simplified.
After distributing \(w\), we get \(4w - w^3\), making the equation \(4w - w^3 = 8 - w^3\).
Polynomial equations can sometimes be solved by isolating terms or factoring.
Here, we added \(w^3\) to both sides to cancel out terms and simplify the equation to \(4w = 8\).
This reduced form is much easier to solve.

Substitution

Substitution is a method where we replace a variable with a known value to check our solution.
It's often used to verify our answers in algebraic equations.
In the provided solution, we substituted \(w=2\) back into the original equation \(2(4 - 2^2) = 8 - 2^3\) to ensure it holds true.
This helps confirm that our solution is correct.
Substitution is a crucial step to avoid errors and validate our results. Always remember, correctly checking your work can give you confidence in your final answer.