Problem 40 - Algebra and Trigonometry

Question

Solve: \(x^{3}=72 x\)

Step-by-Step Solution

Answer

The solutions are \(x = 0\), \(x = 6\sqrt{2}\), and \(x = -6\sqrt{2}\).

Step 1: Write down the equation

Start with the given equation: \[ x^3 = 72x \]

Step 2: Move all terms to one side

To solve for \(x\), first rearrange the equation so that one side equals zero: \[ x^3 - 72x = 0 \]

Step 3: Factor out the common term

Notice that \(x\) is a common factor in both terms. Factor \(x\) out from the equation: \[ x(x^2 - 72) = 0 \]

Step 4: Solve for the factors set to zero

Set each factor equal to zero and solve for \(x\): \[ x = 0 \] and \[ x^2 - 72 = 0 \]

Step 5: Solve the quadratic equation

Solve the remaining quadratic equation \(x^2 - 72 = 0\) by isolating \(x^2\): \[ x^2 = 72 \] Then, take the square root of both sides: \[ x = \pm\sqrt{72} \] This simplifies to: \[ x = \pm 6\sqrt{2} \]

Step 6: Combine all solutions

Collect all solutions found: \[ x = 0, x = 6\sqrt{2}, x = -6\sqrt{2} \]

Key Concepts

Factoring Polynomials

Factoring polynomials is a method used to simplify polynomial expressions by expressing them as a product of simpler polynomials. In our example, we started with the polynomial equation \(x^3 = 72x\).
By rearranging it to \(x^3 - 72x = 0\), we could then look for common factors.
This allowed us to factor out \(x\), obtaining \(x(x^2 - 72) = 0\).
Factoring plays a crucial role in simplifying and solving polynomials because it breaks down complex expressions into manageable parts.
It's especially useful in solving polynomial equations because setting each factor to zero gives us simpler equations to solve.

Quadratic Equations

Quadratic equations are a type of polynomial equation of the second degree, typically written in the form \(ax^2 + bx + c = 0\).
In our example, after factoring out \(x\), we were left with the quadratic equation \(x^2 - 72 = 0\).
To solve this, we isolated \(x^2\) to get \(x^2 = 72\).
Taking the square root of both sides gave us two potential solutions: \(x = \pm\sqrt{72}\).
This simplifies to \(x = \pm 6\sqrt{2}\).
Understanding quadratic equations is essential because they appear frequently in various areas of mathematics, physics, engineering, and other fields.
The methods used to solve quadratic equations, such as factoring, completing the square, and using the quadratic formula, are powerful tools for finding solutions to a wide range of problems.

Solutions of Equations

Finding the solutions of equations involves determining the values of the variable that make the equation true.
In our problem, the solutions were derived from setting each factor to zero:

• From \(x = 0\)
• From \(x^2 - 72 = 0\), which gives \(x = \pm 6\sqrt{2}\)

The concept of solving equations is fundamental to mathematics because it allows us to find unknown values that satisfy given conditions.
It's important to check each solution to ensure it satisfies the original equation.
Understanding how to solve different types of equations, whether they are linear, quadratic, or higher-degree polynomials, is a key skill that forms the basis for more advanced mathematical problem-solving.