How to Find the Solution of Cubic Equation

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In a cubic equation, the highest exponent is 3, the equation has 3 solutions/roots, and the equation itself takes the form ${\displaystyle ax^{3}+bx^{2}+cx+d=0}$ . While cubics look intimidating and can in fact be quite difficult to solve, using the right approach (and a good amount of foundational knowledge) can tame even the trickiest cubics. You can try, among other options, using the quadratic formula, finding integer solutions, or identifying discriminants.

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   Use zero and the quadratic answers as your cubic's answers. While quadratic equations have two solutions, cubics have three. You already have two of these — they're the answers you found for the "quadratic" portion of the problem in parentheses. In cases where your equation is eligible for this "factoring" method of solving, your third answer will always be ${\displaystyle 0}$ .^[6]

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   Plug in the integers manually for a simpler but possibly time-consuming approach. Once you have your list of values, you can find the integer answers to your cubic equation by quickly plugging each integer in manually and finding which ones equal ${\displaystyle 0}$ . For instance, if you plug in ${\displaystyle 1}$ , you get:^[10]

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   Calculate the discriminant of zero using the proper formula. The discriminant approach to finding a cubic equation's solution requires some complicated math, but if you follow the process carefully, you'll find that it's an invaluable tool for figuring out those cubic equations that are hard to crack any other way. To start, find ${\displaystyle \Delta _{0}}$ (the discriminant of zero), the first of several important quantities we'll need, by plugging the appropriate values into the formula ${\displaystyle \Delta _{0}=b^{2}-3ac}$ .^[13]

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• Question

  How would I solve xy+z+z^3=1?

  

  That equation has numerous answers because you've got three variables. To get one answer for three variables you need three equations. One possible answer would be x=1, y=-1, z=1 => (1)(-1)+1+1^3=1.

• Question

  The question is: if 3 consecutive even numbers are multiplied and the result would be 960. What are those numbers and how did you did with the step?

  

  Elvis Kiprotich

  Community Answer

  Solve the equation using the discriminant approach you will get three values of x. X=8, X=-7+(-1i)√71, X=-7+i√71. Easy from here, you pick the real value of x, that's 8 and your three numbers were 8, 10 and 12.

• Question

  Can you give a particular formula for solving cubic equations?

  

  Yes, but it's highly impractical to memorize or even use: http://www.math.vanderbilt.edu/~schectex/courses/cubic/

• Question

  How can there be a square root of -3?

  

  In the ordinary sense, there is no such thing as the square root of a negative number. However, mathematicians have invented the "imaginary" number known as "i", which is defined as the square root of negative 1. The square root of -3 is equal to "i" multiplied by the square root of 3, or 1.73 i.

• Question

  What's the product of Alpha Beta Gamma Delta if they are the roots of the polynomial?

  

  Hemant Dikshit

  Community Answer

  It is the constant term of the polynomial. This is because Minus Alpha x Minus Beta x Minus Gamma x Minus Delta = Plus Alpha Beta Gamma Delta.

• Question

  The link you gave above for a particular formula for solving cubic equations only seems to give one solution; how does one use that formula to get all 3 potential solutions?

  

  Eric Shen

  Community Answer

  By the Fundamental Theorem of Algebra, we have ax^3 + bx^2 + cx + d, which can be expressed as a(x-r)(x-s)(x-t). WLOG let the equation give r. Then, simply divide the cubic by (x-r) and we get a quadratic whose roots are the remaining two roots.

• Question

  What is the solution to 6y3 + 4y2 -5y = 2?

  

  ayodeji oyenaike

  Community Answer

  Multiply and collect: 6y3 + 4y2 - 5y = 2, therefore 21y = 2. The answer is y = 2 / 21.

• Question

  Can anyone factorize x^3+4x-2?

  

  Yes. It is possible. All you need to do is use the factoring approach, only first you must add 2 to both sides. After, you can factor it to (x) (x^2 + 4) = 2. Divide both sides by x to get x^2 + 4 = 2/x. Subtract 4 from both sides to get x^2 = 2/x - 4. Square root both sides to get x = ±√(2x - 4).

• Question

  If N3+N=2x, how do I find N?

  

  First simplify the equation to 4N = 2x. Then divide each side by 4 to get N = (2x)/(4).

• Question

  How to solve A^3 -A = 60?

  

  Before trying advanced methods like the cubic formula, do a quick check for rational roots -- you might get lucky. Here the Rational Roots Theorem implies than any rational roots must be integer divisors of 60. A little trial and error then reveals 4^3 - 4 = 64-4 = 60, so A=4 is a solution. If you require all real and complex solutions, use the known solution to factor A^3 - A - 60 = (A-4)(A^2 + 4A + 15). The quadratic factor has no real roots, but its two complex solutions can be found via the quadratic formula.

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Article SummaryX

To solve a cubic equation, start by determining if your equation has a constant. If it doesn't, factor an x out and use the quadratic formula to solve the remaining quadratic equation. If it does have a constant, you won't be able to use the quadratic formula. Instead, find all of the factors of a and d in the equation and then divide the factors of a by the factors of d. Then, plug each answer into the equation to see which one equals 0. Whichever integer equals 0 is your answer. Read on to learn how to solve a cubic equation using a discriminant approach!

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How to Find the Solution of Cubic Equation

Source: https://www.wikihow.com/Solve-a-Cubic-Equation

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