Factor the expression on the left side of each equation. Then solve the equation. 64x^3-1=0
To factor the expression 64x^3 - 1, we can use the difference of cubes formula, which states that a^3 - b^3 = (a - b)(a^2 + ab + b^2).
First, let's identify the variables: a = 4x and b = 1.
Now we can apply the difference of cubes formula:
64x^3 - 1 = (4x)^3 - 1 = (4x - 1)((4x)^2 + (4x)(1) + 1^2)
= (4x - 1)(16x^2 + 4x + 1)
Now that we have factored the expression, we can set it equal to zero and solve for x:
(4x - 1)(16x^2 + 4x + 1) = 0
Now we have two factors, (4x - 1) and (16x^2 + 4x + 1), multiplied together equaling zero.
To find the solutions, we set each factor equal to zero and solve for x:
First factor: 4x - 1 = 0
Adding 1 to both sides: 4x = 1
Dividing both sides by 4: x = 1/4
Second factor: 16x^2 + 4x + 1 = 0
This quadratic equation can be factored as follows:
(4x + 1)(4x + 1) = 0
Simplifying, we have:
(4x + 1)^2 = 0
Taking the square root of both sides, we get:
4x + 1 = 0
Subtracting 1 from both sides, we have:
4x = -1
Dividing both sides by 4, we get:
x = -1/4
So the solutions to the equation 64x^3 - 1 = 0 are x = 1/4 and x = -1/4.