Solve For \[$ X \$\].$\[ 4 = \sqrt[3]{6x - 2} \\]A. \[$ X = 11 \$\] B. \[$ X = 1 \$\] C. \[$ X = 0 \$\] D. \[$ X = 13 \$\]

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Introduction

In this article, we will delve into the world of cubic equations and learn how to solve for x in a given equation. Cubic equations are a type of polynomial equation of degree three, which means the highest power of the variable (in this case, x) is three. These equations can be challenging to solve, but with the right techniques and strategies, we can find the value of x that satisfies the equation.

Understanding the Equation

The given equation is:

4=6x−234 = \sqrt[3]{6x - 2}

This equation involves a cubic root, which means we need to isolate the variable x and get rid of the cubic root. To do this, we will use the property of cubic roots, which states that if a=b3a = \sqrt[3]{b}, then a3=ba^3 = b.

Step 1: Cube Both Sides

To get rid of the cubic root, we will cube both sides of the equation:

43=(6x−23)34^3 = (\sqrt[3]{6x - 2})^3

This simplifies to:

64=6x−264 = 6x - 2

Step 2: Add 2 to Both Sides

Next, we will add 2 to both sides of the equation to isolate the term with x:

64+2=6x−2+264 + 2 = 6x - 2 + 2

This simplifies to:

66=6x66 = 6x

Step 3: Divide Both Sides by 6

Finally, we will divide both sides of the equation by 6 to solve for x:

666=6x6\frac{66}{6} = \frac{6x}{6}

This simplifies to:

11=x11 = x

Conclusion

Therefore, the value of x that satisfies the equation is x=11x = 11. This is the correct answer among the options provided.

Answer Key

  • A. x=11x = 11
  • B. x=1x = 1
  • C. x=0x = 0
  • D. x=13x = 13

Why is this the Correct Answer?

The correct answer is x=11x = 11 because it is the only value that satisfies the equation. When we substitute x=11x = 11 into the original equation, we get:

4=6(11)−234 = \sqrt[3]{6(11) - 2}

4=66−234 = \sqrt[3]{66 - 2}

4=6434 = \sqrt[3]{64}

4=44 = 4

This shows that x=11x = 11 is indeed the correct solution to the equation.

Tips and Tricks

When solving cubic equations, it's essential to remember the following tips and tricks:

  • Use the property of cubic roots to get rid of the cubic root.
  • Cube both sides of the equation to eliminate the cubic root.
  • Add or subtract the same value to both sides of the equation to isolate the term with x.
  • Divide both sides of the equation by the coefficient of x to solve for x.

Introduction

In our previous article, we learned how to solve for x in a cubic equation. Now, let's put our knowledge to the test with a Q&A session. We'll cover some common questions and scenarios that may arise when working with cubic equations.

Q: What is a cubic equation?

A: A cubic equation is a type of polynomial equation of degree three, which means the highest power of the variable (in this case, x) is three. It can be written in the form:

ax3+bx2+cx+d=0ax^3 + bx^2 + cx + d = 0

where a, b, c, and d are constants.

Q: How do I solve a cubic equation?

A: To solve a cubic equation, you can use the following steps:

  1. Use the property of cubic roots to get rid of the cubic root.
  2. Cube both sides of the equation to eliminate the cubic root.
  3. Add or subtract the same value to both sides of the equation to isolate the term with x.
  4. Divide both sides of the equation by the coefficient of x to solve for x.

Q: What is the property of cubic roots?

A: The property of cubic roots states that if a=b3a = \sqrt[3]{b}, then a3=ba^3 = b. This means that if you have a cubic root, you can cube both sides of the equation to eliminate the cubic root.

Q: How do I cube both sides of an equation?

A: To cube both sides of an equation, you simply multiply both sides of the equation by itself three times. For example, if you have the equation:

a=b3a = \sqrt[3]{b}

you can cube both sides by multiplying both sides by itself three times:

a3=(b3)3a^3 = (\sqrt[3]{b})^3

This simplifies to:

a3=ba^3 = b

Q: What if I have a cubic equation with a coefficient of x?

A: If you have a cubic equation with a coefficient of x, you can divide both sides of the equation by the coefficient of x to solve for x. For example, if you have the equation:

6x3+2x2+3x+1=06x^3 + 2x^2 + 3x + 1 = 0

you can divide both sides of the equation by 6 to get:

x3+13x2+12x+16=0x^3 + \frac{1}{3}x^2 + \frac{1}{2}x + \frac{1}{6} = 0

Q: Can I use algebraic methods to solve cubic equations?

A: Yes, you can use algebraic methods to solve cubic equations. One common method is to use the rational root theorem, which states that if a rational number p/q is a root of the polynomial, then p must be a factor of the constant term and q must be a factor of the leading coefficient.

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

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

  • Not using the property of cubic roots to eliminate the cubic root.
  • Not cubing both sides of the equation to eliminate the cubic root.
  • Not adding or subtracting the same value to both sides of the equation to isolate the term with x.
  • Not dividing both sides of the equation by the coefficient of x to solve for x.

Conclusion

Solving cubic equations can be challenging, but with the right techniques and strategies, you can find the value of x that satisfies the equation. Remember to use the property of cubic roots, cube both sides of the equation, add or subtract the same value to both sides of the equation, and divide both sides of the equation by the coefficient of x to solve for x.