How to Find the Cube Root of xxx is equal to 2 Using Algebra

20 December 2023Last Updated: 20 December 2023

142 2 minutes read

The cube root of a number is a number that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because 2 * 2 * 2 = 8. The cube root of a number can be written as the number raised to the power of one-third. For example, the cube root of 8 can be written as 8^(1/3).

But what if we want to find the cube root of a number that is not a perfect cube, such as 2? How can we use algebra to solve this problem? In this blog post, we will show you how to find the cube root of 2 using a quadratic equation. We will also answer some frequently asked questions about cube roots and algebra.

What is a Quadratic Equation?

A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is a variable. A quadratic equation can have two, one, or no real solutions, depending on the value of the discriminant, which is b^2 – 4ac.

One of the methods to solve a quadratic equation is by using the quadratic formula, which is:

x = (-b ± √(b^2 – 4ac)) / (2a)

The quadratic formula gives the two possible values of x that satisfy the equation. However, not all quadratic equations can be solved by using the quadratic formula. Some quadratic equations may have complex solutions, which involve imaginary numbers, such as √(-1).

How to Find the Cube Root of 2 Using a Quadratic Equation?

To find the cube root of 2 using a quadratic equation, we need to follow these steps:

Step 1: Rewrite the equation xxx = 2 as x^3 – 2 = 0. This is a quadratic equation in terms of x^3, where a = 1, b = 0, and c = -2.
Step 2: Use the quadratic formula to find the value of x^3 that satisfies the equation. We get:

x^3 = (-0 ± √(0^2 – 4 * 1 * (-2))) / (2 * 1)

x^3 = (± √(8)) / 2

x^3 = ± √(2) / √(2)

Step 3: Take the cube root of both sides of the equation to find the value of x. We get:

x = (√(2) / √(2))^(1/3) or x = (-√(2) / √(2))^(1/3)

x = (√(2))^(1/3) / (√(2))^(1/3) or x = (-√(2))^(1/3) / (√(2))^(1/3)

x = 1 or x = -1

Step 4: Check the solutions by plugging them back into the original equation. We get:

xxx = 2

1 * 1 * 1 = 2 or -1 * -1 * -1 = 2

1 = 2 or -1 = 2

Both solutions are false, which means that the equation has no real solutions. Therefore, the cube root of 2 is not a real number but a complex number.

Conclusion

We have shown how to find the cube root of 2 using a quadratic equation. We have also learned that the cube root of 2 is not a real number but a complex number. This means that the cube root of 2 cannot be represented by a point on the number line but by a point on the complex plane.

20 December 2023Last Updated: 20 December 2023

142 2 minutes read