Question

Solve the following equation by factoring?
24x^3 - 81 = 0.

Answer 1

To solve the equation 24x^3 - 81 = 0 by factoring, we can follow these steps:

1. First, we recognize that 24x^3 - 81 is a difference of cubes. This means we can factor it using the formula a^3 - b^3 = (a - b)(a^2 + ab + b^2).

2. Rewrite the equation in the form of a difference of cubes:

24x^3 - 81 = (2x)^3 - 3^4

3. Apply the formula for the difference of cubes:

(2x)^3 - 3^4 = (2x - 3)((2x)^2 + (2x)(3) + 3^2)

4. Simplify the expression within the parentheses:

(2x - 3)(4x^2 + 6x + 9)

5. Set each factor equal to zero and solve for x:

2x - 3 = 0 or 4x^2 + 6x + 9 = 0

6. Solve the first equation:

2x - 3 = 0

2x = 3

x = 3/2

7. Solve the second equation by factoring or using the quadratic formula:

4x^2 + 6x + 9 = 0

However, the quadratic equation does not factor easily in this case, so we'll use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this equation, a = 4, b = 6, and c = 9.

x = (-6 ± √(6^2 - 4(4)(9))) / (2(4))

x = (-6 ± √(36 - 144)) / 8

x = (-6 ± √(-108)) / 8

Since the square root of a negative number is not a real number, the equation has no real solutions.

Therefore, the solutions to the equation 24x^3 - 81 = 0, obtained by factoring, are x = 3/2 and there are no real solutions for the quadratic equation.

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Answer 2

The equation 24x^3 - 81 = 0 is solved by recognizing it is a difference of two cubes and factoring it as such. The resulting factors are (2x - 3)(4x^2 + 6x + 9) = 0. The only real solution is x = 3/2, as the other factor does not yield real roots.

Step-by-step explanation:

To solve the equation 24x^3 - 81 = 0 by factoring, we can recognize that we have a difference of two cubes, as both 24 and 81 are perfect cubes of 2^3 and 3^4, respectively, and x^3 is obviously a cube. The difference of two cubes factorization is expressed as a^3 - b^3 = (a - b)(a^2 + ab + b^2). So, let's factor our equation using this identity:

First, we rewrite the equation as (2^3 \u00d7 x^3) - (3^4) = (2x)^3 - 3^4.

Then we apply the difference of two cubes formula:

(2x - 3)((2x)^2 + 2x \u00d7 3 + 3^2) = 0

(2x - 3)(4x^2 + 6x + 9) = 0

Now we have the product of two factors equal to zero, which means we can set each factor individually equal to zero to solve for x:

1. 2x - 3 = 0 --> 2x = 3 --> x = 3/2
2. 4x^2 + 6x + 9 cannot be factored further and does not have real solutions because it's the sum of squares.

Therefore, the real solution to the equation 24x^3 - 81 = 0 is x = 3/2.