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Chapter 7: Problem 11

Solve each equation. $$\frac{1}{2} x^{3}+4=0$$

Short Answer

Expert verified
x = -2

Step by step solution

01

Isolate the Term with the Variable

Subtract 4 from both sides of the equation to isolate the term with the variable: $$\frac{1}{2} x^{3} = -4$$
02

Eliminate the Fraction

Multiply both sides of the equation by 2 to get rid of the fraction: $$x^{3} = -8$$
03

Take the Cube Root of Both Sides

Take the cube root of both sides of the equation: $$x = \root{3}{-8}$$ Simplify the cube root:

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Key Concepts

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

isolate variable
Isolating the variable is the first step in solving many algebraic equations. The goal here is to get the variable expression alone on one side of the equation.
For the given equation, \(\frac{1}{2}x^{3} + 4 = 0\), we need to isolate the term \( \frac{1}{2} x^{3}\).
This is done by performing the opposite operation of what is currently being done to the variable.
In this case, we subtract 4 from both sides of the equation:
\(\frac{1}{2} x^{3} = -4\).
Remember: whatever operation you perform on one side of the equation, you must do to the other side as well. This keeps the equation balanced.
eliminate fraction
Fractions can make equations look complicated, but there鈥檚 a straightforward way to eliminate them.
To eliminate a fraction coefficient, you multiply both sides of the equation by the denominator of the fraction.
In our cubic equation, the fraction is \( \frac{1}{2} \).
So, we multiply both sides by 2 to cancel out the fraction:
\[ 2 \times \frac{1}{2} x^{3} = 2 \times -4 \].
The left side simplifies to \( x^{3} \), and the right side becomes \( -8 \), resulting in:
\( x^{3} = -8 \).
This crucial step often makes the equation much simpler to solve.
cube root
To solve for \( x \) in \( x^{3} = -8 \, we need to find the cube root of both sides.
Taking the cube root reverses the process of cubing a number.
In our example, we will compute:
\[ x = \root{3}{-8} \].
The cube root of \( -8 \) is \( -2 \), since \( (-2) \times (-2) \times (-2) = -8 \).
Hence, \ ( x = -2 \) is the solution to the equation.
Taking cube roots is an essential operation when dealing with cubic equations. Always remember that cube roots of negative numbers are also negative, which is different from square roots.

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