Problem 34 - Algebra 2

Question

Solve each equation. $$ 27 x^{3}+1=0 $$

Step-by-Step Solution

Answer

\(x = -\frac{1}{3}\).

Step 1: Rewrite the equation

The equation given is \(27x^3 + 1 = 0\). First, we need to isolate the term with \(x^3\). Here, we can rewrite the equation as \(27x^3 = -1\).

Step 2: Simplify the equation

Divide each side of the equation by 27 to solve for \(x^3\). This gives us \(x^3 = -\frac{1}{27}\).

Step 3: Solve for x

Find the cube root of both sides of the equation. The cube root of a number \(a\) is a number \(b\) such that \(b^3 = a\). Therefore, taking the cube root of both sides of the equation we get \(x = \sqrt[3]{-\frac{1}{27}}\).

Step 4: Calculate the cube root

The cube root of \(-\frac{1}{27}\) is \(-\frac{1}{3}\) because \(\left(-\frac{1}{3}\right)^3 = -\frac{1}{27}\). Thus, \(x = -\frac{1}{3}\).

Key Concepts

Cubic Equations

Cubic equations are polynomials of degree three. This means they take the form:
\[ax^3 + bx^2 + cx + d = 0\]where \(a, b, c,\) and \(d\) are constants, and \(a eq 0\). In algebra, solving cubic equations involves finding the value(s) of \(x\) that satisfy the equation. Since the highest power is three, it means there can be up to three real roots.

When you work with cubic equations, it's common to see them in operations involving factoring, simplifying, or using the cubic formula for more complex equations. They are a step up from quadratic equations and are prevalent in various real-world applications where a relationship isn't linear or quadratic. Understanding how to manipulate and solve these equations is key for many applications in physics, engineering, and other sciences.

Isolation of Variables

Isolation of variables refers to the process of manipulating an equation to get a specific variable on one side by itself. This technique is crucial in simplifying equations to make them solvable.

To isolate a variable, you often need to perform operations such as adding, subtracting, multiplying, or dividing by certain terms. For instance, in the equation \(27x^3 + 1 = 0\), our goal was to isolate \(x^3\):

• First, eliminate any constant terms on the same side as the variable. In this case, subtract 1 to get \(27x^3 = -1\).
• Then, divide each term by the coefficient of \(x^3\), which is 27 here. This gives \(x^3 = -\frac{1}{27}\), leaving \(x^3\) isolated.
  

By isolating variables, you simplify the equation and bring it into a form that can be more easily solved or analyzed.

Cube Roots

The cube root of a number is a value that produces that number when multiplied by itself twice. Mathematically, for a given number \(a\), its cube root is a number \(b\), such that \(b^3 = a\). The notation for the cube root is \(\sqrt[3]{a}\).

In solving our equation from earlier, \(x^3 = -\frac{1}{27},\) the next step is finding the cube root:

• Recognize the cube root operation is the inverse of cubing. By applying it, you "undo" the cube to find \(x\).
• Calculate the cube root of \(-\frac{1}{27}\). Since \(\left(-\frac{1}{3}\right)^3 = -\frac{1}{27}\), the cube root of \(-\frac{1}{27}\) is \(-\frac{1}{3}\).
  

Understanding cube roots is essential when dealing with cubic equations, as it provides a direct way to find a solution for \(x\) when \(x^3\) is known. Cube roots extend beyond just negative numbers, as they can be calculated for both positive and negative real numbers.