Answer:
To find the real solutions for the equation 2x^3 + 4 = x^2 + 8x, we need to first rearrange the equation and then factor it. Here's the step-by-step process:
1. Start by moving all terms to one side of the equation to set it equal to zero:
2x^3 + 4 - (x^2 + 8x) = 0
2x^3 - x^2 - 8x + 4 = 0
2. Now, let's try to factor the equation. We can start by factoring out the greatest common factor (GCF), which is 2:
2(x^3 - 1/2x^2 - 4x + 2) = 0
3. Now, focus on factoring the cubic polynomial inside the parentheses. To do this, you can try the rational root theorem to find a rational root (if it exists). The rational root theorem suggests that any rational root of the polynomial must be of the form ±(p/q), where p is a divisor of the constant term (2 in this case) and q is a divisor of the leading coefficient (1 in this case).
The divisors of 2 are ±1 and ±2, and the divisors of 1 are ±1, so the possible rational roots are ±1, ±2.
Now, you can try these potential roots to see if they make the polynomial equal to zero. Starting with x = 1:
f(1) = (1^3 - 1/2 * 1^2 - 4 * 1 + 2) = (1 - 1/2 - 4 + 2) = -5/2
So, x = 1 is not a root.
Next, try x = -1:
f(-1) = (-1^3 - 1/2 * (-1)^2 - 4 * (-1) + 2) = (-1 + 1/2 + 4 + 2) = 7/2
So, x = -1 is not a root.
Now, try x = 2:
f(2) = (2^3 - 1/2 * 2^2 - 4 * 2 + 2) = (8 - 2 - 8 + 2) = 0
So, x = 2 is a root.
Therefore, we have found one real root, which is x = 2.
4. To find the remaining roots, you can perform polynomial division to divide the cubic polynomial by (x - 2) since we already know that x = 2 is a root. This will give you a quadratic polynomial, which you can solve using the quadratic formula or by factoring.
I'll leave the polynomial division and quadratic factoring/solving as an exercise for you, but keep in mind that you have already found one real root, which is x = 2, and there may be additional real roots or complex roots for the quadratic polynomial you obtain after division.
Explanation: