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I have been hours trying to solve this cubic equation any help? $2x^3-7x^2+1.6x+7=0$

Majd
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    Ask WA. – lhf Oct 11 '19 at 13:37
  • There is no easy formula, I would suggest numerical methods. – Peter Oct 11 '19 at 13:37
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    Why should you expect solving it would have been easy? Absolute worst case scenario if you want an exact solution, you can use cardano's formulas, but hardly anyone ever memorizes that. Otherwise, you can use methods to approximate the solutions such as how a calculator like wolfram alpha might have done. – JMoravitz Oct 11 '19 at 13:37
  • Well maybe i need to rewrite my question correctly, they actually asked for how much answers for this equation, so maybe this one will be easier to answer – Majd Oct 11 '19 at 13:49
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    If you need to know on number of real solutions, then it's easy to calculate (the sign of) discriminant (https://en.wikipedia.org/wiki/Discriminant#Degree_3) – Oleg567 Oct 11 '19 at 13:56
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    For that, what tools do you have available? Have you learned calculus and derivatives yet? From your function's derivative $6x^2-14x+1.6$ you should be able to find the zeroes of which will give you the locations of the local minima and local maxima of your original function. You see that the local minima is negative and the local maxima is positive, which will imply that your original equation has three distinct real roots. – JMoravitz Oct 11 '19 at 13:56
  • I still cant understand how to find how much real answers this cubic equation have? Any details – Majd Oct 11 '19 at 14:09
  • Downvote because you didnt add any content to it how would be we able to help when we dont upto which level or what methods you are aware of this was an easy question if you would have known calculus – Akshaj Bansal Oct 11 '19 at 14:31

2 Answers2

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If the qubic equation $$ax^3+bx^2+cx+d=0\tag{1}$$ is given, then one of easiest ways to know the number of its real roots (since the number of complex roots is $3$ anyway) is to calculate special value called Discriminant (Discriminant (Wiki)).

It is some sort of "indicators" of the number of real roots for polynomials of small degree.

For quadratic equations $ax^2+bx+c=0$ there exists simple formula: $D=b^2-4ac$.

For cubic equation of the form $(1)$ we have a bit longer formula:

$$ D = b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2+18abcd. \tag{2} $$

If $D=0$, then at least two solutions are equal.

If $D<0$, then there is $1$ real solution of eq. ($1$) (and two complex conjugate ones).

If $D>0$, then there are three distinct real solutions.

When evaluate expression $(2)$, one will obtain positive value, so?

Oleg567
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  • actually when i calculated it , its gave me an negative value , so its have 1 real and two complex – Majd Oct 11 '19 at 17:34
  • @Majd: oops, it should be $>0$: since $a=2,;; b=-7,;; c=1.6,;;d=7$, we have $$D = 49\cdot 2.56 - 4\cdot 2\cdot 4.096 + 4\cdot 343\cdot 7 - 27\cdot 4\cdot 49-18\cdot 2\cdot 7\cdot 1.6\cdot 7 = \ 125.44 - 32.768 + 9604 - 5292 - 2822.4 = 1582.272 > 0.$$ – Oleg567 Oct 11 '19 at 18:35
  • Wolframalpha checking link (approximate value): https://www.wolframalpha.com/input/?i=Discriminant%282x%5E3-7x%5E2%2B1.6x%2B7%29 – Oleg567 Oct 11 '19 at 18:40
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To solve this question, you can use the cubic formula, it is a very long and complicated formula, I hope this helps! enter image description here

Max Chan
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    Surely there should be at least two $\pm$ signs in the formula since there are three complex roots? – Clement Yung Oct 11 '19 at 13:43
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    Since the cubic has three real roots, this is the casus irreducibilis and requires complex numbers to solve these square roots. Or trigonometric functions. – lhf Oct 11 '19 at 13:44
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    I will emphasize that hardly anyone ever uses this without a computer doing it for them, and using the related formula for quartics is even worse. It is also worth pointing out that no such formula can possibly exist for quintics or above. – JMoravitz Oct 11 '19 at 13:44
  • Well maybe i need to rewrite my question correctly, they actually asked for how much answers for this equation, so maybe this one will be easier to answer? – Majd Oct 11 '19 at 13:49
  • I still cant understand how to find how much real answers this cubic equation have? Any details – Majd Oct 11 '19 at 14:08