Show that the equation 3x^2 - x^3 + 3 = 0 can be rearranged to give:
x = 3 + 3
----
x^2
x^3 =
Divide by ? = x = 3 + 3
----
x^2
Using Xn+1 = 3 + 3 with x0 = 3.2
----
x^2n
find the values of x1 x2 and x3?
What to the values of x1 x2 and x3 represent?
The represent estimates of a _____ to the equation?
Just re-arrange it ....
3x^2 - x^3 + 3 = 0
-x^3 = -3x^2 - 3
x^3 = 3x^2 + 3
divide both sides by x^2
x = 3 + 3/x^2
let x = 3.2
RS = 3+ 3/10.24 = 3.29296875
sub that into the RS
RS = 3 + 3/(3.2929...)^2 = 3.276659...
new RS = 3.279420685
new RS = 3.278950402
new RS = 3.279030424
new RS = 3.279016806
new RS = 3.279019123
new RS = 3.279018729
new RS = 3.279018796
ok, once more
new RS = 3.279018784 , good enough?, we could get closer!
Wolfram shows that this is the only real solution
http://www.wolframalpha.com/input/?i=y+%3D+3x%5E2+-+x%5E3+%2B+3
If you need the two complex roots, use synthetic division to reduce your cubic to a quadratic and use the formula
answer confirmed by
http://www.wolframalpha.com/input/?i=solve+3x%5E2+-+x%5E3+%2B+3+%3D+0
click on "exact forms" and then even on "more digits" to show how close we got
Well, I could solve this equation for you, but I think it would be more fun to juggle some numbers instead! Here we go:
Let's start with the original equation: 3x^2 - x^3 + 3 = 0.
To rearrange it, we'll add x^3 to both sides: 3x^2 + 3 = x^3.
Now, let's divide both sides by x^2: (3x^2 + 3) / x^2 = x^3 / x^2.
This simplifies to: 3 + 3 / x^2 = x.
Aha! We have successfully rearranged the equation to x = 3 + 3 / x^2.
Now, moving on to the values of x1, x2, and x3. We can use the recursive formula Xn+1 = 3 + 3 / x^2n with x0 = 3.2.
So, let's calculate:
x1 = 3 + 3 / (3.2)^2,
x2 = 3 + 3 / (x1)^2,
x3 = 3 + 3 / (x2)^2.
Now, the values of x1, x2, and x3 represent estimates of a... drumroll please... the root to the equation! Did you get it? Yes, they are approximations of the solutions to the equation 3x^2 - x^3 + 3 = 0.
And now, my juggling act is complete! Let me know if you need any more mathematical tricks or if you have any other questions.
To rearrange the equation 3x^2 - x^3 + 3 = 0 to the form x = 3 + 3/x^2, we need to isolate the x term on one side of the equation. Here are the step-by-step instructions:
1. Start with the equation: 3x^2 - x^3 + 3 = 0.
2. Subtract 3 from both sides to isolate the terms with x: 3x^2 - x^3 = -3.
3. Move the x^3 term to the left side: -x^3 = -3 - 3x^2.
4. Multiply both sides by -1 to make the coefficient of x^3 positive: x^3 = 3 + 3x^2.
5. Divide both sides by x^2 to obtain the desired form: x^3 / x^2 = (3 + 3x^2) / x^2.
6. Simplify the equation: x = 3 + 3 / x^2.
Now, let's solve for x using the iteration formula Xn+1 = 3 + 3 / Xn^2, with X0 = 3.2:
1. Start with X0 = 3.2.
2. Plug the value of X0 into the iteration formula to find X1: X1 = 3 + 3 / (3.2)^2.
3. Calculate X1: X1 = 3 + 3 / 10.24 = 3 + 0.293 = 3.293.
4. Using X1 as the new value, plug it into the iteration formula to find X2: X2 = 3 + 3 / (3.293)^2.
5. Calculate X2: X2 = 3 + 3 / 10.860049 = 3 + 0.2759 = 3.2759.
6. Using X2 as the new value, plug it into the iteration formula to find X3: X3 = 3 + 3 / (3.2759)^2.
7. Calculate X3: X3 = 3 + 3 / 10.776376 = 3 + 0.2777 = 3.2777.
Therefore, the values of x1, x2, and x3 are approximately 3.293, 3.2759, and 3.2777, respectively.
The values x1, x2, and x3 represent estimates of a root (or a solution) to the equation 3x^2 - x^3 + 3 = 0.
To rearrange the equation 3x^2 - x^3 + 3 = 0 and express x in terms of x^2, we can start by moving the terms around:
3x^2 - x^3 + 3 = 0
Rearranging the terms, we have:
x^3 - 3x^2 + 3 = 0
Factoring out an x, we get:
x(x^2 - 3x + 3) = 0
Now, we can divide both sides of the equation by x to isolate (x^2 - 3x + 3):
x^2 - 3x + 3 = 0
Now, we want to solve for x in terms of x^2. To do this, we can use the equation:
x^3 = 3 + 3/x^2
From the rearranged equation, we can see that x = 3 + (3/x^2). This gives us an iterative equation:
Xn+1 = 3 + (3/Xn^2)
Using this iteration method, we can find the values of x1, x2, and x3. Starting with an initial value of x0 = 3.2:
x1 = 3 + (3/3.2^2) = 3.234375
x2 = 3 + (3/3.234375^2) = 3.234143
x3 = 3 + (3/3.234143^2) = 3.234143
The values of x1, x2, and x3 represent estimates of a root (or solution) to the equation 3x^2 - x^3 + 3 = 0. Each iteration improves the estimate closer to the actual solution.