Problem 47 Solve. $$ x^{3}+1=0 $$... [FREE SOLUTION]

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Chapter 6: Problem 47

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

Short Answer

Expert verified

x = -1

Step by step solution

- Understand the Equation

The equation given is a cubic equation, specifically: \[x^3 + 1 = 0\]. Our goal is to solve for the value of $x$.

- Isolate the Cubic Term

Rearrange the equation to isolate the cubic term by subtracting 1 from both sides:\[x^3 = -1\].

- Solve for x

To solve for $x$, take the cube root of both sides of the equation:\[x = \sqrt[3]{-1}\]. The cube root of $-1$ is $-1$ since $(-1)^3 = -1$. Therefore, \[x = -1\].

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Key Concepts

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

solving cubic equations

Cubic equations are equations of the form $ax^3 + bx^2 + cx + d = 0$.
They can be more complex to solve compared to linear or quadratic equations.
We specifically looked at a simpler cubic equation:
$x^3 + 1 = 0$.
This involves finding values of $x$ such that $x$ to the power of three equals a given constant.
Let's break it down step by step.
Initially, isolate the cubic term by getting rid of any additional constants.
Then, solve for $x$ by using appropriate algebraic methods.
Finally, verify if the solution works by substituting it back into the original equation.

cube roots

Taking the cube root is crucial in solving cubic equations.
The cube root of a number $a$ is a number $b$ such that $b^3 = a$.
For positive and negative values, the cube root can be a bit tricky.
In the example,
we isolated the cubic term: $x^3 = -1$.
The next step is to take the cube root of both sides.
This gives us $x = \sqrt[3]{-1}$.
Since the cube root of \[-1\] is \[-1\], our solution for this cubic equation is \[x = -1\].

isolating terms

Isolating terms is a key step in solving equations.
To isolate a term means to get it by itself on one side of the equation.
This simplifies the equation and makes it easier to solve.
In our example $x^3 + 1 = 0$, we first isolated $x^3$ by subtracting 1 from both sides:\[x^3 = -1\].
This leaves us with a simpler expression that can be further simplified.

algebraic solutions

Algebraic solutions involve using algebraic methods to solve equations. In the given exercise, we used several algebraic methods:
- Subtracting constants from both sides to isolate terms.
- Taking cube roots to find the value of $x$.
Such steps streamline solving the problem.
After isolating $x^3$ to get $x^3 = -1$,
when the cube root is taken, we resolve the equation as $x = -1$.
Using these algebraic steps often provides a systematic way to find solutions to various types of equations.

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91影视

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

Short Answer

Step by step solution

- Understand the Equation

- Isolate the Cubic Term

- Solve for x

Key Concepts

solving cubic equations

cube roots

isolating terms

algebraic solutions

One App. One Place for Learning.

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