What are the real and complex solutions of the polynomial equation?
x3 – 64 =0
The polynomial equation x^3 - 64 = 0 can be factored as (x - 4)(x^2 + 4x + 16) = 0.
The real solution is x = 4.
The complex solutions can be found by solving the quadratic equation x^2 + 4x + 16 = 0.
Using the quadratic formula, where a = 1, b = 4, and c = 16:
x = (-b ± √(b^2 - 4ac))/(2a)
= (-4 ± √(4^2 - 4(1)(16)))/(2(1))
= (-4 ± √(16 - 64))/2
= (-4 ± √(-48))/2
= (-4 ± 4√3i)/2
= -2 ± 2√3i
Therefore, the complex solutions are x = -2 + 2√3i and x = -2 - 2√3i.
are you sure?
Apologies for the mistake in my previous response. The correct factorization of x^3 - 64 = 0 is (x - 4)(x^2 + 4x + 16) = 0.
The real solution is x = 4.
To find the complex solutions, we need to solve the quadratic equation x^2 + 4x + 16 = 0.
Using the quadratic formula, where a = 1, b = 4, and c = 16:
x = (-b ± √(b^2 - 4ac))/(2a)
= (-4 ± √(4^2 - 4(1)(16)))/(2(1))
= (-4 ± √(16 - 64))/2
= (-4 ± √(-48))/2
At this point, we can simplify further.√(-48) can be expressed as 4√(-3) since √3 has been already factored out as a common factor of 4.
= (-4 ± 4√(-3))/2
= -2 ± 2√(-3)
So, the complex solutions are x = -2 + 2√(-3) and x = -2 - 2√(-3) in terms of the imaginary unit √(-1).
To find the real and complex solutions of the polynomial equation x^3 - 64 = 0, we can follow these steps:
Step 1: Factor the equation (if possible)
In this case, we can use the formula for the difference of cubes:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
By applying this formula, we can rewrite x^3 - 64 as (x - 4)(x^2 + 4x + 16).
Step 2: Solve for real solutions
Setting each factor equal to zero and solving for x, we have:
x - 4 = 0 --> x = 4
So, the real solution to the equation is x = 4.
Step 3: Solve for complex solutions
For the complex solutions, we consider the quadratic factor, x^2 + 4x + 16, and solve it using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/2a
For this equation, a = 1, b = 4, and c = 16. Plugging in these values, we get:
x = (-4 ± √(4^2 - 4 * 1 * 16)) / (2 * 1)
= (-4 ± √(-48)) / 2
= (-4 ± √(48)i) / 2
= (-4 ± 4√(3)i) / 2
Simplifying further, we have two complex solutions:
x = (-4 + 4√(3)i) / 2
= -2 + 2√(3)i
x = (-4 - 4√(3)i) / 2
= -2 - 2√(3)i
So, the complex solutions to the equation are x = -2 + 2√(3)i and x = -2 - 2√(3)i.