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Chapter 11: Problem 38

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

Short Answer

Expert verified
x = -1

Step by step solution

01

Move the constant to the other side

Begin by isolating the term involving the variable. Subtract 1 from both sides of the equation to get: \[ x^3 + 1 - 1 = 0 - 1 \] This simplifies to: \[ x^3 = -1 \]
02

Take the cube root of both sides

To solve for \( x \), take the cube root of both sides of the equation: \[ \frac{d}{dx}( x^3 ) = \frac{d}{dx}( -1 ) \] This results in: \[ x = \frac{-1}{3} \] However, we are looking for the cube root: \[ x = \sqrt[3]{-1} \]
03

Simplify the cube root

Evaluate the cube root of \( -1 \). By definition, the cube root of \( -1 \) is \( -1 \): \[ x = -1 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Variable
When solving equations, especially cubic ones, the first step is to isolate the term that contains the variable. This simplifies the equation and makes it easier to solve. In our problem, we start with: \[x^{3}+1=0\].
To isolate the variable term, we need to move the constant to the other side of the equation. Do this by subtracting 1 from both sides:
\[ x^3 + 1 - 1 = 0 - 1\]
By doing so, we simplify our equation to:
\[ x^3 = - 1 \]
Now, the variable term \(x^3\) stands alone on one side.
Cube Root
The next crucial concept is understanding the cube root, which is the inverse operation of cubing a number. Since our equation is \(x^3 = -1\), we need to find what number, when cubed, gives \(-1\).
To do this, apply the cube root to both sides of the equation:
\[\text{cube root}( x^3 ) = \text{cube root}( -1 )\]
This simplifies to:
\[ x = \root 3 \bigg( -1 \bigg) \]
A helpful tip to remember is that the cube root of a negative number will also be negative. This differs from square roots, where only positive solutions are possible.
Simplification Steps
The final step is to simplify the cube root to find the value of our variable \(x\). In this problem, we determine:
\( x = \root 3 \bigg( -1 \bigg) \)
Knowing basic cube roots is key here. For \( x = \root 3 \bigg( -1 \bigg) \), we recognize that the cube root of \(-1\) is simply \(-1\):
\( x = -1 \)
This means that \(-1\) is our solution.

To recap the steps:
  • Isolate the term with the variable by moving constants to the other side
  • Apply the cube root to both sides of the equation
  • Simplify the cube root to find the value of your variable

Applying these steps allows you to solve cubic equations efficiently.

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Most popular questions from this chapter

To prepare for Section \(11.2,\) review evaluating expressions and simplifying radical expressions (Sections \(1.8,10.3\) and \(10.8)\) Simplify. \([10.3],[10.8]\) $$ \sqrt{200} $$

For each of the following, write the equation of the parabola that has the shape of \(f(x)=2 x^{2}\) or \(g(x)=-2 x^{2}\) and has a maximum value or \(a\) minimum value at the specified point. Minimum: \((2,0)\)

Solve. $$ (x+1)^{2}=-9 $$

Replace the blanks in each equation with constants to complete the square and form a true equation. \(x^{2}+8 x+\)_____\(=(x+\)_____\()^{2}\)

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