Find All Solutions to the Cubic Equation x³+1=0

Q: Why doesn't this cubic equation have three solutions like I learned?

Great question! While cubic equations can have up to 3 solutions, they don't always have 3 real solutions. This equation $ x³ + 1 = 0 $ has one real solution and two complex solutions.

Q: How do I know when to take a cube root versus a square root?

Look at the highest power of your variable! If you have $ x³ $, take the cube root. If you have $ x² $, take the square root. Match the root to the exponent.

Q: Can I get a negative number under a cube root?

Yes! Unlike square roots, cube roots of negative numbers are perfectly fine. $ \sqrt[3]{-8} = -2 $ because $ (-2)³ = -8 $.

Q: What's the difference between real and complex solutions?

Real solutions are regular numbers you can plot on a number line. Complex solutions involve the imaginary unit i and can't be plotted on a regular number line.

Q: How do I check if my cube root is correct?

Cube your answer! If $ x = -1 $, then $ (-1)³ = -1 $. Always verify: does $ x³ = -1 $? ✓

Cubic Equations with Real Solutions

How many solutions does the equation have?

$x^3+1=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's find the value of X.

00:08 First, we need to isolate X on one side of the equation.

00:18 Then, take the cube root to determine X.

00:24 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

How many solutions does the equation have?

$x^3+1=0$

Step-by-step solution

In the given equation:

$x^3+1=0$ The simplest and fastest way to find the number of its solutions,

will be simply to solve it, we will do this by moving terms to isolate the unknown, then we will take the cube root of both sides of the equation, while remembering that an odd root preserves the sign of the expression inside the root (meaning - the minus sign can be taken out of an odd root):

$x^3+1=0 \\ x^3=-1\hspace{6pt}\text{/}\sqrt[3]{\hspace{4pt}}\\ \downarrow\\ \sqrt[3]{x^3}=\sqrt[3]{-1}\\ x=-\sqrt[3]{1}\\ \boxed{x=-1}$ meaning the given equation has a single solution,

therefore the correct answer is answer A.

Final Answer

A solution

Key Points to Remember

Essential concepts to master this topic

Rule: Odd roots preserve signs, so $\sqrt[3]{-8} = -2$
Technique: Isolate x³ first: $x³ = -1$ , then take cube root of both sides
Check: Substitute x = -1: $(-1)³ + 1 = -1 + 1 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Thinking cubic equations always have three real solutions
Don't assume all cubic equations have three real solutions just because the degree is 3 = wrong count! While cubics can have up to 3 solutions, some may be complex (imaginary). Always solve the equation to find how many real solutions exist.

Practice Quiz

Test your knowledge with interactive questions

Solve the following expression:

$$ x^2-1=0 $$

FAQ

Everything you need to know about this question

Why doesn't this cubic equation have three solutions like I learned?

Great question! While cubic equations can have up to 3 solutions, they don't always have 3 real solutions. This equation $x³ + 1 = 0$ has one real solution and two complex solutions.

How do I know when to take a cube root versus a square root?

Look at the highest power of your variable! If you have $x³$ , take the cube root. If you have $x²$ , take the square root. Match the root to the exponent.

Can I get a negative number under a cube root?

Yes! Unlike square roots, cube roots of negative numbers are perfectly fine. $\sqrt[3]{-8} = -2$ because $(-2)³ = -8$ .

What's the difference between real and complex solutions?

Real solutions are regular numbers you can plot on a number line. Complex solutions involve the imaginary unit i and can't be plotted on a regular number line.

How do I check if my cube root is correct?

Cube your answer! If $x = -1$ , then $(-1)³ = -1$ . Always verify: does $x³ = -1$ ? ✓

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