John, I have been working your problems for over a year so far. I investigated several other sites that work similar to yours and found them sorely lacking in detail. My original study was all the math's up to and including two semesters of Calculus, but that was 55+ years ago. Currently could not Calculus my way out of a wet paper sack. I am finding your content and explanations most excellent. Thanks.

Good lesson! 👍

Thanks for the review. Haven’t used this method in years!

Cubic equations with real coefficients always have at least one real root. Often it is possible to find a real root by inspection, as in this case. x=1 is a real root. If a real root cannot easily be obtained by inspection, Newton-Raphson iteration may be used. Let f(x)=4x^3-13x^2+11x-2 and f'(x)=12x^2-26x+11. Then make a plot to estimate a root and set x to that value. Then iterate x←x-f(x)/f'(x). Once the root has been determined, divide by (x-1) in this case to obtain 4x^2-9x+2=0. The quadratic solution yields x=(9±7)/8=2 or 1/4.

interesting and fun. thanks.

I'm going with 4x^3 - 13x^2 + 11x = 2. If that's incorrect, someone please let me know.

1:49 Brute forced this one again. Waiting for the explanation of the math. 4:03 bummer, I missed the .25 part of the answer :(

1, 2, 1/4

Personally I think the best way to solve these equations is to graph them.

The answer is 1/4 not -1/4. When you displayed the solutions at the end you had negative 1/4.

@terry_willis

5 ай бұрын

I noticed that too. What happened? It appeared he caught his mistake and then back tracked and kept -1/4. Which is it?

x=1 by inspection. Took 15 secs.

I got 2, but not the others

Did not watch yet. But already feel like I will disagree with “must know”, for this problem. Maybe for higher orders it becomes a must 🤓. Because it exists, I would apply the formula for cube polynomials. Not nearly as simple as for squares, but it exists, though I think it is not taught, ever…. 4th powers, I recall, can be solved by applying what I expect is part of the vid to show. 5th powers, and above, are proven to just never have a ‘simple’ (closed form) formula to follow. We’ll see if I get to eat my words …

X-1is a factor. Divide to get a trinomial and factorize the trinomial ans solve for x for less than 20 minutes. Wasting time