Still having problems solving this...
2x(2x + 1)^2 = 312
I started with:
2x(2x+1)(2x+1)-312 = 0
2x(4x^2+4x+1)-312 = 0
8x^3+8x^2+2x -312 = 0
Now what?
If you don't have to solve for x, then then you have the final answer!
If you have to find x: then you have to use the group method.
(8x^3+8x^2)+ (2x-312)
8x^2(x + 1) + 2(x-156)
Since there doesn't seem to be any like terms after factor. The answer may have well been what you had above. Someone on here may second this.
I do have to solve for x...
Is this a cubic polynomial???
Still not sure.
8x^3+8x^2+2x -312 = 0
Now what?
divide by 2:
4x^3+ 4x^2+ x - 156 = 0
Try the Rational Roots theorem. If there is a root of the form p/q with p and q integers that don't have common factors, then p must be a divisor of 156 and q a divisor of 4. I don't see a candidate that works...
If you draw the graph of this function (it helps to calculate the derivative to find the local maximum and minimum), you see that this function has only one root at approximately x = 3. Using a high precision numerical algorithm I find that the root is at:
x = 3.06633690898....
This is certainly not a rational root, because the only possible rational roots would have to become an integer after multiplication by 4
It is possible to exactly calculate the roots of cubic equation. If we divide the equation by 4 we get:
x^3+ x^2 + 1/4x - 39 = 0
First you substitute:
x = y - 1/3
and work out the polynomial as a function of y. This has the effect of eliminating the quadratic term. We get:
y^3 + p y + q = 0 (1)
with:
p = -1/12
q = -4213/108
You solve this equation by comparing this equation to the equation for
(A + B)^3:
(A + B)^3 = A^3 + 3 A^2B + 3 A B^2 + B^3
rearranging gives:
(A + B)^3 - 3AB(A + B)-(A^3 + B^3) = 0
What use is this? Well, suppose you can find A and B such that
3AB = -p (2)
and
A^3 + B^3 = -q (3)
then Y = A + B would satisfy the equation (1)
This should be possible because we need to solve two equations for two unknowns A and B. So, how do we go about solving for A and B? What you do is you take the third power of Eq. (2):
27 A^3 B^3 = -p^3
If you now put A^3 = s and B^3 = t then this means that
s*t = -p^3/27
while Eq. (3) implies that:
s + t = -q
Combining these two equations for
s and t gives you a quadratic equation for these quantities. By extracting the cube root you find A and B,. Add them together and you find y. Subtract 1/3 to find x.
To solve the equation 8x^3 + 8x^2 + 2x - 312 = 0, we can start by trying to factorize it. However, we notice that it doesn't easily factorize using common methods.
Next, we can try the Rational Roots theorem. According to this theorem, if there is a rational root of the form p/q, where p is a divisor of -312 and q is a divisor of 8, then it would be a potential solution. However, after checking all possible combinations, we don't find a rational root that satisfies the equation.
Graphing the equation can provide some insight. By calculating the derivative, we can find the local maximum and minimum points of the graph. From the graph, it appears that there is only one real root, approximately x = 3.
Using a high precision numerical algorithm, we can find the root to be x = 3.06633690898...
Since this solution is not rational, we can try to transform the equation to eliminate the quadratic term. We can substitute x = y - 1/3, which will give us a new equation in terms of y. This substitution eliminates the quadratic term, and we get y^3 + p y + q = 0, where p = -1/12 and q = -4213/108.
Now, we can try to solve this new equation by comparing it to the equation for (A + B)^3: (A + B)^3 = A^3 + 3 A^2B + 3 A B^2 + B^3. By rearranging, we have (A + B)^3 - 3AB(A + B) - (A^3 + B^3) = 0.
To find A and B, we need to solve two equations for these two unknowns. First, we have 3AB = -p, and second, we have A^3 + B^3 = -q. We can raise the polynomial 3AB = -p to the third power to obtain 27A^3B^3 = -p^3.
By letting A^3 = s and B^3 = t, we have s * t = -p^3/27, while A^3 + B^3 = -q. Combining these two equations yields a quadratic equation for s and t. By extracting the cube root, we can find A and B. Adding them together gives us y = A + B. Subtracting 1/3 from y gives us x.