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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 . 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.
Steps
Method 1
Method 1 of 3:Solving Cubic Equations without a Constant

1Check whether your cubic contains a constant (a value). Cubic equations take the form . However, the only essential requirement is , which means the other elements need not be present to have a cubic equation.^{[1] X Research source }
 If your equation does contain a constant (a value), you'll need to use another solving method.
 If , you do not have a cubic equation.^{[2] X Research source }

2Factor an out of the equation. Since your equation doesn't have a constant, every term in the equation has an variable in it. This means that one can be factored out of the equation to simplify it. Do this and rewrite your equation in the form .^{[3] X Research source }
 For example, let's say that your starting cubic equation is
 Factoring a single out of this equation, you get
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3Factor the resulting quadratic equation, if possible. In many cases, you will be able to factor the quadratic equation () that results when you factor the out. For example, if you are given , then you can do the following:^{[4] X Research source }
 Factor out the :
 Factor the quadratic in parentheses:
 Set each of these factors equal to. Your solutions are .

4Solve the portion in parentheses with the quadratic formula if you can’t factor it manually. You can find the values for which this quadratic equation equals by plugging , , and into the quadratic formula (). Do this to find two of the answers to your cubic equation.^{[5] X Research source }
 In the example, plug your , , and values (, , and , respectively) into the quadratic equation as follows:
 Answer 1:
 Answer 2:
 In the example, plug your , , and values (, , and , respectively) into the quadratic equation as follows:

5Use 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 .^{[6] X Research source }
 Factoring your equation into the form splits it into two factors: one factor is the variable on the left, and the other is the quadratic portion in parentheses. If either of these factors equals , the entire equation will equal .
 Thus, the two answers to the quadratic portion in parentheses, which will make that factors equal , are answers to the cubic, as is itself, which will make the left factor equal .
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Method 2
Method 2 of 3:Finding Integer Solutions with Factor Lists

1Ensure your cubic has a constant (a nonzero value). If your equation in the form has a nonzero value for , factoring with the quadratic equation won't work. But don’t worry—you have other options, like the one described here!^{[7] X Research source }
 Take, for example, . In this case, getting a on the right side of the equals sign requires you to add to both sides.
 In the new equation, . Since , you can't use the quadratic equation method.

2Find the factors of and . Start solving the cubic equation by finding the factors of the coefficient of the term (that is, ) and the constant at the end of the equation (that is, ). Remember that factors are the numbers that can multiply together to make another number.^{[8] X Research source }
 For example, since you can make 6 by multiplying and , that means 1, 2, 3, and 6 are the factors of 6.
 In the sample problem, and . The factors of 2 are 1 and 2. The factors of 6 are 1, 2, 3, and 6.

3Divide the factors of by the factors of . Make a list of the values you get by dividing each factor of by each factor of . This will usually result in lots of fractions and a few whole numbers. The integer solutions to your cubic equation will either be one of the whole numbers in this list or the negative of one of these numbers.^{[9] X Research source }
 In the sample equation, taking the factors of (1 and 2) over the factors of (1, 2, 3 and 6) gets this list: , , , , , and . Next, we add the negatives to the list to make it complete: , , , , , , , , , , , and . Your cubic equation's integer solutions are somewhere in this list.

4Plug in the integers manually for a simpler but possibly timeconsuming 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 . For instance, if you plug in , you get:^{[10] X Research source }
 , or , which clearly does not equal . So, move on to the next value on your list.
 If you plug in , you get , which does equal . This means is one of your integer solutions.

5Employ synthetic division for a more complex but likely faster approach. If you don't want to spend the time plugging in values one by one, try a quicker method that involves a technique called synthetic division. Basically, you'll want to synthetically divide your integer values by the original , , , and coefficients in your cubic equation. If you get a remainder of , your value is one of the cubic equation's answers.^{[11] X Research source }
 Synthetic division is a complex topic that’s beyond the scope of describing fully here. However, here's a sample of how to find one of the solutions to your cubic equation with synthetic division:
 1  2 9 13 6
 __ 276
 __ 2 7 6 0
 Since you got a final remainder of , you know that one of your cubic's integer solutions is .
Advertisement  Synthetic division is a complex topic that’s beyond the scope of describing fully here. However, here's a sample of how to find one of the solutions to your cubic equation with synthetic division:
Method 3
Method 3 of 3:Using a Discriminant Approach

1Write out the values of , , , and . For this method you’ll be dealing heavily with the coefficients of the terms in your equation. Record your , , , and terms before you begin so you don't forget what each one is.^{[12] X Research source }
 For the sample equation , write , , , and . Don't forget that when an variable doesn't have a coefficient, it's implicitly assumed that its coefficient is .

2Calculate 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 (the discriminant of zero), the first of several important quantities we'll need, by plugging the appropriate values into the formula .^{[13] X Research source }
 A discriminant is simply a number that gives us information about the roots of a polynomial (you may already know the quadratic discriminant: ).
 In your sample problem, solve as follows:

3Follow up by calculating . The next important quantity you’ll need, (the discriminant of ), requires a little more work, but is found in essentially the same way as . Plug the appropriate values into the formula to get your value for .^{[14] X Research source }
 In the example, solve as follows:
 In the example, solve as follows:

4Calculate: . Next, we'll calculate the discriminant of the cubic from the values of and . In the case of the cubic, if the discriminant is positive, then the equation has three real solutions. If the discriminant is zero, then the equation has either one or two real solutions, and some of those solutions are shared. If it is negative, then the equation has only one solution.^{[15] X Research source }
 A cubic equation always has at least one real solution, because the graph will always cross the xaxis at least once.
 In the example, since both and , finding is relatively easy. Solve as follows:
 , so the equation has one or two answers.

5Calculate: . The last important value we need to calculate is . This important quantity will allow us to finally find our three roots. Solve as normal, substituting and as needed.
 In your example, find as follows:
 In your example, find as follows:

6Calculate the three roots with your variables. The roots (answers) to your cubic equation are given by the formula , where and n is either 1, 2, or 3. Plug in your values as needed to solve — this requires lots of mathematical legwork, but you should receive three viable answers!
 You can solve the example by checking the answer when n is equal to 1, 2, and 3. The answers you get from these tests are the possible answers to the cubic equation — any that give an answer of 0 when plugged into the equation are correct.
 For example, since plugging 1 into gives an answer of 0, 1 is one of the answers to your cubic equation.
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Community Q&A

QuestionHow would I solve xy+z+z^3=1?Community AnswerThat 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.

QuestionThe 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 KiprotichCommunity AnswerSolve 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.

QuestionCan you give a particular formula for solving cubic equations?Community AnswerYes, but it's highly impractical to memorize or even use: http://www.math.vanderbilt.edu/~schectex/courses/cubic/

QuestionHow can there be a square root of 3?DonaganTop AnswererIn 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.

QuestionWhat's the product of Alpha Beta Gamma Delta if they are the roots of the polynomial?Hemant DikshitCommunity AnswerIt 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.

QuestionThe 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 ShenCommunity AnswerBy the Fundamental Theorem of Algebra, we have ax^3 + bx^2 + cx + d, which can be expressed as a(xr)(xs)(xt). WLOG let the equation give r. Then, simply divide the cubic by (xr) and we get a quadratic whose roots are the remaining two roots.

QuestionWhat is the solution to 6y3 + 4y2 5y = 2?ayodeji oyenaikeCommunity AnswerMultiply and collect: 6y3 + 4y2  5y = 2, therefore 21y = 2. The answer is y = 2 / 21.

QuestionCan anyone factorize x^3+4x2?Community AnswerYes. 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).

QuestionIf N3+N=2x, how do I find N?Community AnswerFirst simplify the equation to 4N = 2x. Then divide each side by 4 to get N = (2x)/(4).

QuestionHow to solve A^3 A = 60?Community AnswerBefore 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 = 644 = 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 = (A4)(A^2 + 4A + 15). The quadratic factor has no real roots, but its two complex solutions can be found via the quadratic formula.
Video
Tips
References
 ↑ http://www.mathcentre.ac.uk/resources/uploaded/mctycubicequations20091.pdf
 ↑ https://sciencing.com/solvecubicequations8136094.html
 ↑ https://sciencing.com/solvecubicequations8136094.html
 ↑ http://www.mathcentre.ac.uk/resources/uploaded/mctycubicequations20091.pdf
 ↑ https://www.purplemath.com/modules/quadform.htm
 ↑ https://math.vanderbilt.edu/schectex/courses/cubic/
 ↑ http://www.rasmus.is/uk/t/F/Su52k02.htm
 ↑ http://www.rasmus.is/uk/t/F/Su52k02.htm
 ↑ http://www.rasmus.is/uk/t/F/Su52k02.htm
 ↑ http://www.rasmus.is/uk/t/F/Su52k02.htm
 ↑ http://www.rasmus.is/uk/t/F/Su52k02.htm
 ↑ http://www2.trinity.unimelb.edu.au/~rbroekst/MathX/Cubic%20Formula.pdf
 ↑ http://www2.trinity.unimelb.edu.au/~rbroekst/MathX/Cubic%20Formula.pdf
 ↑ http://www2.trinity.unimelb.edu.au/~rbroekst/MathX/Cubic%20Formula.pdf
 ↑ http://www2.trinity.unimelb.edu.au/~rbroekst/MathX/Cubic%20Formula.pdf
About This Article
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!