Solutions to the Given Equation

Oct 13, 2025 by miodeya.com 42 views

Introduction

Cubic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore the different methods for solving cubic equations, with a focus on the given equation: $s^3 = 8$ . We will also discuss the solutions to this equation and provide a step-by-step guide on how to solve it.

What are Cubic Equations?

A cubic equation is a polynomial equation of degree three, which means that the highest power of the variable (in this case, $s$ ) is three. Cubic equations can be written in the form $ax^3 + bx^2 + cx + d = 0$ , where $a$ , $b$ , $c$ , and $d$ are constants, and $x$ is the variable.

The Given Equation: $s^3 = 8$

The given equation is a simple cubic equation, where the variable $s$ is raised to the power of three and equals eight. To solve this equation, we need to find the value of $s$ that satisfies the equation.

Method 1: Taking the Cube Root

One way to solve the equation $s^3 = 8$ is to take the cube root of both sides. This gives us:

s = \sqrt[3]{8}

To evaluate the cube root, we can use the fact that $2^3 = 8$ . Therefore, we can write:

s = \sqrt[3]{2^3}

Using the property of radicals, we can simplify this expression as:

s = 2

Method 2: Factoring

Another way to solve the equation $s^3 = 8$ is to factor the left-hand side. We can write:

s^3 - 8 = 0

This can be factored as:

(s - 2)(s^2 + 2s + 4) = 0

Setting each factor equal to zero, we get:

s - 2 = 0 \quad \text{or} \quad s^2 + 2s + 4 = 0

Solving the first equation, we get:

s = 2

The second equation is a quadratic equation, which can be solved using the quadratic formula:

s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, $a = 1$ , $b = 2$ , and $c = 4$ . Plugging these values into the formula, we get:

s = \frac{-2 \pm \sqrt{2^2 - 4(1)(4)}}{2(1)}

Simplifying this expression, we get:

s = \frac{-2 \pm \sqrt{-12}}{2}

This is a complex number, which is not a real solution to the equation.

Method 3: Using the Rational Root Theorem

The rational root theorem states that if a rational number $p/q$ is a root of the polynomial $f(x)$ , then $p$ must be a factor of the constant term, and $q$ must be a factor of the leading coefficient. In this case, the constant term is $-8$ , and the leading coefficient is $1$ . Therefore, the possible rational roots are:

\pm 1, \pm 2, \pm 4, \pm 8

We can test each of these values by plugging them into the equation $s^3 = 8$ . We find that:

s = 2

is a solution to the equation.

Method 4: Using the Cubic Formula

The cubic formula is a general method for solving cubic equations. It states that if $a$ , $b$ , and $c$ are the roots of the cubic equation $ax^3 + bx^2 + cx + d = 0$ , then:

x = \frac{-b \pm \sqrt{b^2 - 3ac}}{3a}

In this case, $a = 1$ , $b = 0$ , and $c = 0$ . Plugging these values into the formula, we get:

s = \frac{0 \pm \sqrt{0^2 - 3(1)(0)}}{3(1)}

Simplifying this expression, we get:

s = \frac{0 \pm \sqrt{0}}{3}

This is a complex number, which is not a real solution to the equation.

Conclusion

In this article, we have discussed the different methods for solving cubic equations, with a focus on the given equation: $s^3 = 8$ . We have shown that the solutions to this equation are:

s = 2, s = \frac{3}{8}, s = 1, s = 2.6

We have also provided a step-by-step guide on how to solve the equation using different methods, including taking the cube root, factoring, and using the rational root theorem.

Solutions to the Given Equation

Here are the solutions to the given equation:

$s = 2$

This is a real solution to the equation.

$s = \frac{3}{8}$

This is a real solution to the equation.

$s = 1$

This is a real solution to the equation.

$s = 2.6$

This is a real solution to the equation.

Final Thoughts

Solving cubic equations can be a challenging task, but with the right methods and techniques, it can be done. In this article, we have discussed the different methods for solving cubic equations, with a focus on the given equation: $s^3 = 8$ . We have shown that the solutions to this equation are:

s = 2, s = \frac{3}{8}, s = 1, s = 2.6

Q: What is a cubic equation?

A: A cubic equation is a polynomial equation of degree three, which means that the highest power of the variable (in this case, $s$ ) is three. Cubic equations can be written in the form $ax^3 + bx^2 + cx + d = 0$ , where $a$ , $b$ , $c$ , and $d$ are constants, and $x$ is the variable.

Q: How do I solve a cubic equation?

A: There are several methods for solving cubic equations, including:

Taking the cube root of both sides
Factoring the left-hand side
Using the rational root theorem
Using the cubic formula

Q: What is the cube root method?

A: The cube root method involves taking the cube root of both sides of the equation. This can be written as:

s = \sqrt[3]{a}

where $a$ is the constant term on the right-hand side of the equation.

Q: What is the factoring method?

A: The factoring method involves factoring the left-hand side of the equation. This can be written as:

(s - r)(s^2 + ps + q) = 0

where $r$ , $p$ , and $q$ are constants.

Q: What is the rational root theorem?

A: The rational root theorem states that if a rational number $p/q$ is a root of the polynomial $f(x)$ , then $p$ must be a factor of the constant term, and $q$ must be a factor of the leading coefficient.

Q: What is the cubic formula?

A: The cubic formula is a general method for solving cubic equations. It states that if $a$ , $b$ , and $c$ are the roots of the cubic equation $ax^3 + bx^2 + cx + d = 0$ , then:

x = \frac{-b \pm \sqrt{b^2 - 3ac}}{3a}

Q: How do I choose the right method for solving a cubic equation?

A: The choice of method depends on the specific equation and the desired solution. Some methods may be more efficient or accurate than others, depending on the equation.

Q: What are some common mistakes to avoid when solving cubic equations?

A: Some common mistakes to avoid when solving cubic equations include:

Not checking for extraneous solutions
Not using the correct method for the specific equation
Not simplifying the equation before solving
Not checking the solution for accuracy

Q: How do I check my solution for accuracy?

A: To check your solution for accuracy, you can plug the solution back into the original equation and verify that it is true. You can also use a calculator or computer program to check the solution.

Q: What are some real-world applications of cubic equations?

A: Cubic equations have many real-world applications, including:

Physics: Cubic equations are used to model the motion of objects under the influence of gravity.
Engineering: Cubic equations are used to design and optimize systems, such as bridges and buildings.
Computer Science: Cubic equations are used in computer graphics and game development.

Q: Can cubic equations be solved using technology?

A: Yes, cubic equations can be solved using technology, such as calculators and computer programs. These tools can be used to find the solutions to cubic equations quickly and accurately.

Q: What are some tips for solving cubic equations?

A: Some tips for solving cubic equations include:

Start by simplifying the equation
Use the correct method for the specific equation
Check for extraneous solutions
Use technology to check the solution for accuracy