How to Solve a Cubic Equation
Check whether your cubic contains a constant., Factor an x out of the equation., Use the quadratic formula to solve the portion in parentheses., Use zero and the quadratic answers as your cubic's answers.
Step-by-Step Guide
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Step 1: Check whether your cubic contains a constant.
As noted above, cubic equations take the form ax3 + bx2 + cx + d =
0. b, c, and d can be 0 without affecting whether the equation is cubic or not — this basically means that a cubic equation doesn't have to include all of the bx2, cx, or d terms to be a cubic.
To start using this relatively easy method of solving cubics, check to see whether your equation has a constant (i.e., a d value).
If it doesn't, you can use the quadratic equation to find the answers to the equation after a little mathematical legwork.
If, on the other hand, your equation does contain a constant, you'll need to use another solving method.
See below for alternate approaches. -
Step 2: Factor an x out of the equation.
Since your equation doesn't have a constant, every term in the equation has an x variable in it.
This means that one x can be factored out of the equation to simplify it.
Do this and re-write your equation in the form x(ax2 + bx + c).
For example, let's say that our starting cubic equation is 3x3 +
-2x2 + 14x =
0.
Factoring a single x out of this equation, we get x(3x2 +
-2x + 14) =
0. , You may have noticed that the portion of your new equation contained in parentheses matches the form of a quadratic equation (ax2 + bx + c).
This means that we can find the values for which this quadratic equation equals zero by plugging a, b, and c into the quadratic formula ({-b +/-√ (b2- 4ac)}/2a).
Do this to find two of the answers to you cubic equation.
In our example, we would plug our a, b, and c values (3,
-2, and 14, respectively) into the quadratic equation as follows: {-b +/-√ (b2- 4ac)}/2a {-(-2) +/-√ ((-2)2- 4(3)(14))}/2(3) {2 +/-√ (4
- (12)(14))}/6 {2 +/-√ (4
- (168)}/6 {2 +/-√ (-164)}/6 Answer 1: {2 + √(-164)}/6 {2 +
12.8i}/6 Answer 2: {2
-
12.8i}/6 , While quadratic equations have two solutions, cubics have three.
You already have two of these — they're the answers you found to 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
0.
Congratulations — you've just solved your cubic.
The reason this works has to do with the fundamental fact that any number times zero equals zero.
When you factor your equation into the form x(ax2 + bx + c) = 0, you essentially split it into two "halves": one half is the x variable on the left and the other is the quadratic portion in parentheses.
If either of these "halves" equals zero, the entire equation will.
Thus, the two answers to the quadratic portion in parentheses, which will make that "half" equal zero, are answers to the cubic, as is 0 itself, which will make the left "half" equal zero. -
Step 3: Use the quadratic formula to solve the portion in parentheses.
-
Step 4: Use zero and the quadratic answers as your cubic's answers.
Detailed Guide
As noted above, cubic equations take the form ax3 + bx2 + cx + d =
0. b, c, and d can be 0 without affecting whether the equation is cubic or not — this basically means that a cubic equation doesn't have to include all of the bx2, cx, or d terms to be a cubic.
To start using this relatively easy method of solving cubics, check to see whether your equation has a constant (i.e., a d value).
If it doesn't, you can use the quadratic equation to find the answers to the equation after a little mathematical legwork.
If, on the other hand, your equation does contain a constant, you'll need to use another solving method.
See below for alternate approaches.
Since your equation doesn't have a constant, every term in the equation has an x variable in it.
This means that one x can be factored out of the equation to simplify it.
Do this and re-write your equation in the form x(ax2 + bx + c).
For example, let's say that our starting cubic equation is 3x3 +
-2x2 + 14x =
0.
Factoring a single x out of this equation, we get x(3x2 +
-2x + 14) =
0. , You may have noticed that the portion of your new equation contained in parentheses matches the form of a quadratic equation (ax2 + bx + c).
This means that we can find the values for which this quadratic equation equals zero by plugging a, b, and c into the quadratic formula ({-b +/-√ (b2- 4ac)}/2a).
Do this to find two of the answers to you cubic equation.
In our example, we would plug our a, b, and c values (3,
-2, and 14, respectively) into the quadratic equation as follows: {-b +/-√ (b2- 4ac)}/2a {-(-2) +/-√ ((-2)2- 4(3)(14))}/2(3) {2 +/-√ (4
- (12)(14))}/6 {2 +/-√ (4
- (168)}/6 {2 +/-√ (-164)}/6 Answer 1: {2 + √(-164)}/6 {2 +
12.8i}/6 Answer 2: {2
-
12.8i}/6 , While quadratic equations have two solutions, cubics have three.
You already have two of these — they're the answers you found to 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
0.
Congratulations — you've just solved your cubic.
The reason this works has to do with the fundamental fact that any number times zero equals zero.
When you factor your equation into the form x(ax2 + bx + c) = 0, you essentially split it into two "halves": one half is the x variable on the left and the other is the quadratic portion in parentheses.
If either of these "halves" equals zero, the entire equation will.
Thus, the two answers to the quadratic portion in parentheses, which will make that "half" equal zero, are answers to the cubic, as is 0 itself, which will make the left "half" equal zero.
About the Author
Patricia Morgan
Creates helpful guides on lifestyle to inspire and educate readers.
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