Solve \[$(v+6)^3 - 48 = 0\$\] Where \[$v\$\] Is A Real Number. Write Your Answer In Simplified Radical Form. If There Is More Than One Solution, Separate Them With Commas.$\[ V = \\]No Solution

Solving the Cubic Equation: $(v+6)^3 - 48 = 0$

In this article, we will delve into solving a cubic equation of the form $(v+6)^3 - 48 = 0$, where $v$ is a real number. Our goal is to find the value of $v$ in simplified radical form. We will explore the steps involved in solving this equation and provide the solution in the required format.

Understanding the Equation

The given equation is a cubic equation, which means it involves a variable raised to the power of 3. The equation is $(v+6)^3 - 48 = 0$. To solve this equation, we need to isolate the variable $v$.

Step 1: Expand the Cubic Term

The first step in solving this equation is to expand the cubic term $(v+6)^3$. Using the binomial expansion formula, we get:

$$(v+6)^3 = v^3 + 3v^2(6) + 3v(6)^2 + 6^3$$

Simplifying the expression, we get:

$$(v+6)^3 = v^3 + 18v^2 + 108v + 216$$

Step 2: Rewrite the Equation

Now that we have expanded the cubic term, we can rewrite the original equation as:

$$v^3 + 18v^2 + 108v + 216 - 48 = 0$$

Simplifying the equation, we get:

$$v^3 + 18v^2 + 108v + 168 = 0$$

Step 3: Factor the Equation

The next step is to factor the equation. We can start by factoring out the greatest common factor (GCF) of the terms. In this case, the GCF is 4. Factoring out 4, we get:

$$4(v^3 + 18v^2 + 27v + 42) = 0$$

Step 4: Solve for $v$

Now that we have factored the equation, we can set each factor equal to 0 and solve for $v$. We get:

$$v^3 + 18v^2 + 27v + 42 = 0$$

This is a cubic equation, and solving it analytically can be challenging. However, we can try to find the rational roots of the equation using the Rational Root Theorem.

Rational Root Theorem

The Rational Root Theorem states that if a rational number $p/q$ is a root of the polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0$, where $p$ and $q$ are integers and $q$ is nonzero, then $p$ must be a factor of the constant term $a_0$, and $q$ must be a factor of the leading coefficient $a_n$.

In this case, the constant term is 42, and the leading coefficient is 1. The factors of 42 are $\pm 1, \pm 2, \pm 3, \pm 6, \pm 7, \pm 14, \pm 21, \pm 42$. Since the leading coefficient is 1, the possible rational roots are $\pm 1, \pm 2, \pm 3, \pm 6, \pm 7, \pm 14, \pm 21, \pm 42$.

Testing Rational Roots

We can test each of these possible rational roots by substituting them into the equation and checking if the equation is satisfied. After testing, we find that $v = -6$ is a root of the equation.

Factoring the Cubic Equation

Since we have found one root, we can factor the cubic equation as:

$$(v+6)(v^2-6v+7) = 0$$

Solving the Quadratic Equation

The quadratic equation $v^2-6v+7 = 0$ can be solved using the quadratic formula:

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

In this case, $a = 1$, $b = -6$, and $c = 7$. Plugging these values into the formula, we get:

$$v = \frac{6 \pm \sqrt{(-6)^2-4(1)(7)}}{2(1)}$$

Simplifying the expression, we get:

$$v = \frac{6 \pm \sqrt{36-28}}{2}$$

$$v = \frac{6 \pm \sqrt{8}}{2}$$

$$v = \frac{6 \pm 2\sqrt{2}}{2}$$

$$v = 3 \pm \sqrt{2}$$

Conclusion

In conclusion, we have solved the cubic equation $(v+6)^3 - 48 = 0$ and found the value of $v$ in simplified radical form. The solutions are $v = -6$ and $v = 3 \pm \sqrt{2}$.

Final Answer

The final answer is: $\boxed{3 + \sqrt{2}, 3 - \sqrt{2}, -6}$
Solving the Cubic Equation: $(v+6)^3 - 48 = 0$ - Q&A

In our previous article, we solved the cubic equation $(v+6)^3 - 48 = 0$ and found the value of $v$ in simplified radical form. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q: What is the significance of the cubic equation?

A: The cubic equation $(v+6)^3 - 48 = 0$ is a fundamental equation in mathematics, and its solution has various applications in fields such as physics, engineering, and computer science. The equation represents a cubic function, which is a polynomial function of degree 3.

Q: How do I expand the cubic term?

A: To expand the cubic term $(v+6)^3$, you can use the binomial expansion formula:

$$(v+6)^3 = v^3 + 3v^2(6) + 3v(6)^2 + 6^3$$

Simplifying the expression, you get:

$$(v+6)^3 = v^3 + 18v^2 + 108v + 216$$

Q: What is the Rational Root Theorem?

A: The Rational Root Theorem is a theorem in algebra that states that if a rational number $p/q$ is a root of the polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0$, where $p$ and $q$ are integers and $q$ is nonzero, then $p$ must be a factor of the constant term $a_0$, and $q$ must be a factor of the leading coefficient $a_n$.

Q: How do I test rational roots?

A: To test rational roots, you can substitute each possible rational root into the equation and check if the equation is satisfied. In this case, we found that $v = -6$ is a root of the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The formula is:

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

Q: How do I solve the quadratic equation?

A: To solve the quadratic equation $v^2-6v+7 = 0$, you can use the quadratic formula:

$$v = \frac{6 \pm \sqrt{(-6)^2-4(1)(7)}}{2(1)}$$

Simplifying the expression, you get:

$$v = \frac{6 \pm \sqrt{36-28}}{2}$$

$$v = \frac{6 \pm \sqrt{8}}{2}$$

$$v = \frac{6 \pm 2\sqrt{2}}{2}$$

$$v = 3 \pm \sqrt{2}$$

Q: What are the solutions to the cubic equation?

A: The solutions to the cubic equation $(v+6)^3 - 48 = 0$ are $v = -6$ and $v = 3 \pm \sqrt{2}$.

Conclusion

In conclusion, we have provided a Q&A section to help clarify any doubts and provide additional information on the topic of solving the cubic equation $(v+6)^3 - 48 = 0$. We hope this article has been helpful in understanding the concept of solving cubic equations and the quadratic formula.

Final Answer

The final answer is: $\boxed{3 + \sqrt{2}, 3 - \sqrt{2}, -6}$