Factor and solve: 4x^3-32=0
Please so work so I can finish on my own with an example problem. Thanks! <3
To factor and solve the equation 4x^3 - 32 = 0, we can follow these steps:
Step 1: Factor out the greatest common factor (GCF) from the equation, if possible.
In this case, the GCF is 4. So, we can factor out 4 to get:
4(x^3 - 8) = 0
Step 2: Identify and factor any special products or patterns, if applicable.
In this case, we have a difference of cubes. The difference of cubes formula states that a^3 - b^3 = (a - b)(a^2 + ab + b^2). We can recognize that x^3 - 8 is a difference of cubes, where a = x and b = 2. Therefore, we can use the formula to factor it as:
x^3 - 8 = (x - 2)(x^2 + 2x + 4)
Step 3: Set each factor equal to zero and solve for x.
Setting each factor equal to zero, we have two equations to solve:
1) x - 2 = 0
Solving for x, we have:
x = 2
2) x^2 + 2x + 4 = 0
This is a quadratic equation that cannot be factored further. We can solve it using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values of a, b, and c from the equation, we get:
x = (-2 ± √(2^2 - 4(1)(4))) / (2(1))
Simplifying further:
x = (-2 ± √(4 - 16)) / 2
x = (-2 ± √(-12)) / 2
x = (-2 ± 2i√3) / 2
x = -1 ± i√3
Therefore, the solutions to the equation 4x^3 - 32 = 0 are:
x = 2, -1 + i√3, -1 - i√3
By following these steps, you can try solving a similar problem on your own.