Which Of The Following Represents All Of The Solutions To The Equation Below?$x^3 = 512$A. $x = 8$ B. $x = \pm 8$ C. $x = \pm 23$ D. $x = 23$

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Introduction

In mathematics, solving equations is a fundamental concept that helps us understand various mathematical operations and relationships. One of the most common types of equations is the cubic equation, which is a polynomial equation of degree three. In this article, we will focus on solving the cubic equation x3=512x^3 = 512 and explore all possible solutions.

Understanding the Equation

The given equation is x3=512x^3 = 512. To solve this equation, we need to find the value of xx that satisfies the equation. The equation is a cubic equation because the highest power of the variable xx is three.

Solving the Equation

To solve the equation x3=512x^3 = 512, we can start by taking the cube root of both sides of the equation. This will give us the value of xx that satisfies the equation.

x=5123x = \sqrt[3]{512}

Finding the Cube Root of 512

The cube root of 512 can be found by using a calculator or by using the prime factorization method. The prime factorization of 512 is 292^9. Therefore, the cube root of 512 is 232^3, which is equal to 8.

x=5123=8x = \sqrt[3]{512} = 8

Checking for Other Solutions

However, we need to check if there are any other solutions to the equation. To do this, we can rewrite the equation as x3−512=0x^3 - 512 = 0. This is a cubic equation, and it can have up to three solutions.

Using the Factor Theorem

The factor theorem states that if aa is a root of the polynomial f(x)f(x), then (x−a)(x - a) is a factor of f(x)f(x). We can use this theorem to find the other solutions to the equation.

Factoring the Cubic Equation

We can factor the cubic equation x3−512=0x^3 - 512 = 0 as follows:

x3−512=(x−8)(x2+8x+64)=0x^3 - 512 = (x - 8)(x^2 + 8x + 64) = 0

Finding the Other Solutions

The quadratic factor x2+8x+64x^2 + 8x + 64 can be factored further as follows:

x2+8x+64=(x+4)2+64=0x^2 + 8x + 64 = (x + 4)^2 + 64 = 0

However, this quadratic factor does not have any real solutions. Therefore, the only real solution to the equation is x=8x = 8.

Checking for Complex Solutions

However, we need to check if there are any complex solutions to the equation. To do this, we can use the quadratic formula to solve the quadratic factor x2+8x+64=0x^2 + 8x + 64 = 0.

Using the Quadratic Formula

The quadratic formula states that the solutions to the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 are given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

We can use this formula to solve the quadratic factor x2+8x+64=0x^2 + 8x + 64 = 0.

Solving the Quadratic Factor

Using the quadratic formula, we get:

x=−8±82−4(1)(64)2(1)x = \frac{-8 \pm \sqrt{8^2 - 4(1)(64)}}{2(1)}

Simplifying, we get:

x=−8±64−2562x = \frac{-8 \pm \sqrt{64 - 256}}{2}

x=−8±−1922x = \frac{-8 \pm \sqrt{-192}}{2}

Finding the Complex Solutions

The complex solutions to the equation are given by:

x=−8±−1922x = \frac{-8 \pm \sqrt{-192}}{2}

Simplifying, we get:

x=−8±8−32x = \frac{-8 \pm 8\sqrt{-3}}{2}

x=−4±4−3x = -4 \pm 4\sqrt{-3}

Conclusion

In conclusion, the solutions to the equation x3=512x^3 = 512 are x=8x = 8 and x=−4±4−3x = -4 \pm 4\sqrt{-3}. Therefore, the correct answer is:

  • A. x=8x = 8 is partially correct, but it does not represent all of the solutions to the equation.
  • B. x=±8x = \pm 8 is correct, as it represents all of the solutions to the equation.
  • C. x=±23x = \pm 23 is incorrect, as it does not represent any of the solutions to the equation.
  • D. x=23x = 23 is incorrect, as it does not represent any of the solutions to the equation.

Q: What is the cube root of 512?

A: The cube root of 512 is 8.

Q: How do I find the cube root of a number?

A: You can find the cube root of a number by using a calculator or by using the prime factorization method. The prime factorization of 512 is 2^9, so the cube root of 512 is 2^3, which is equal to 8.

Q: What is the difference between the cube root and the square root?

A: The cube root and the square root are both roots of a number, but they are different. The cube root of a number is the number that, when multiplied by itself three times, gives the original number. The square root of a number is the number that, when multiplied by itself, gives the original number.

Q: Can I use the cube root to solve equations?

A: Yes, you can use the cube root to solve equations. For example, if you have the equation x^3 = 512, you can take the cube root of both sides of the equation to get x = 8.

Q: What is the factor theorem?

A: The factor theorem states that if a is a root of the polynomial f(x), then (x - a) is a factor of f(x). This means that if you know the roots of a polynomial, you can use the factor theorem to find the factors of the polynomial.

Q: How do I use the factor theorem to find the factors of a polynomial?

A: To use the factor theorem to find the factors of a polynomial, you need to know the roots of the polynomial. Once you know the roots, you can use the factor theorem to find the factors of the polynomial. For example, if you know that the root of the polynomial x^3 - 512 is 8, you can use the factor theorem to find the factors of the polynomial.

Q: What is the difference between a real solution and a complex solution?

A: A real solution is a solution to an equation that is a real number. A complex solution is a solution to an equation that is a complex number. In the case of the equation x^3 = 512, the real solution is x = 8, and the complex solutions are x = -4 ± 4√(-3).

Q: Can I use the quadratic formula to solve quadratic equations?

A: Yes, you can use the quadratic formula to solve quadratic equations. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula to solve quadratic equations?

A: To use the quadratic formula to solve quadratic equations, you need to know the coefficients of the quadratic equation. Once you know the coefficients, you can plug them into the quadratic formula to find the solutions to the equation.

Q: What is the difference between a quadratic equation and a cubic equation?

A: A quadratic equation is a polynomial equation of degree two, while a cubic equation is a polynomial equation of degree three. In the case of the equation x^3 = 512, the equation is a cubic equation because the highest power of the variable x is three.

Q: Can I use the quadratic formula to solve cubic equations?

A: No, you cannot use the quadratic formula to solve cubic equations. The quadratic formula is used to solve quadratic equations, while the cubic formula is used to solve cubic equations.

Q: What is the cubic formula?

A: The cubic formula is a formula that is used to solve cubic equations. The cubic formula is x = (-q/2 + √(q^2/4 + p^3/27)) / 3, where p and q are the coefficients of the cubic equation.

Q: How do I use the cubic formula to solve cubic equations?

A: To use the cubic formula to solve cubic equations, you need to know the coefficients of the cubic equation. Once you know the coefficients, you can plug them into the cubic formula to find the solutions to the equation.

Conclusion

In conclusion, the cube root of 512 is 8, and the solutions to the equation x^3 = 512 are x = 8 and x = -4 ± 4√(-3). The factor theorem can be used to find the factors of a polynomial, and the quadratic formula can be used to solve quadratic equations. The cubic formula can be used to solve cubic equations.