Difference between revisions of "Cubic Equation"
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− | A cubic equation is an equation of the form: | + | A '''cubic equation''' is an equation of the form: |
<math>ax^3 + bx^2 + cx + d = 0</math>. | <math>ax^3 + bx^2 + cx + d = 0</math>. | ||
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Divide both sides by a: <math>x^3 + \frac{b}{a}x^2 + \frac{c}{a}x + \frac{d}{a}</math>. | Divide both sides by a: <math>x^3 + \frac{b}{a}x^2 + \frac{c}{a}x + \frac{d}{a}</math>. | ||
− | Now we change the [[coefficient]] of <math>x^2</math> to <math>0</math> (e.g. change it to a depressed cubic). We do this by [[substitution|substituting]] <math> | + | Now we change the [[coefficient]] of <math>x^2</math> to <math>0</math> (e.g. change it to a depressed cubic). We do this by [[substitution|substituting]] <math>x = y - \frac{b}{3a}</math>, giving: |
<math>\left(y - \frac{b}{3a}\right)^3 + \frac{b}{a}\left(y - \frac{b}{3a}\right)^2 + \frac{c}{a}\left(y - \frac{b}{3a}\right) + \frac{d}{a} = 0</math> | <math>\left(y - \frac{b}{3a}\right)^3 + \frac{b}{a}\left(y - \frac{b}{3a}\right)^2 + \frac{c}{a}\left(y - \frac{b}{3a}\right) + \frac{d}{a} = 0</math> | ||
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<math>u - v - (3\sqrt[3]{uv} - p)(\sqrt[3]{u} - \sqrt[3]{v}) = -q</math> | <math>u - v - (3\sqrt[3]{uv} - p)(\sqrt[3]{u} - \sqrt[3]{v}) = -q</math> | ||
− | We want that last term to equal <math>0</math>, so we can set <math>3\sqrt[3]{uv} - p = 0</math>. (We can't use <math>\sqrt[3]{u} - \sqrt[3]{v} = 0</math>, because then <math>y = 0</math>, which is not necessarily true.) Solving this equation gives us <math>uv = \frac{p^3}{27}</math>. If <math>uv = \frac{p^3}{27}</math>, then <math>v - u = q</math>. We now have a system of equations: | + | We want that last term to equal <math>0</math>, so we can set <math>3\sqrt[3]{uv} - p = 0</math>. (We can't use <math>\sqrt[3]{u} - \sqrt[3]{v} = 0</math>, because then <math>y = 0</math>, which is not necessarily true.) Solving this equation gives us <math>uv = \frac{p^3}{27}</math>. If <math>uv = \frac{p^3}{27}</math>, then <math>v - u = q</math>. We now have a [[system of equations]]: |
<math>\begin{cases} uv = \frac{p^3}{27} \\ v - u = q \end{cases}</math>. | <math>\begin{cases} uv = \frac{p^3}{27} \\ v - u = q \end{cases}</math>. | ||
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The depressed cubic is of the form <math>y^3 + \left(\frac{3ac - b^2}{3a^2}\right)y + \left(\frac{2b^3 - 9abc + 27a^2d}{27a^3}\right) = 0</math>. | The depressed cubic is of the form <math>y^3 + \left(\frac{3ac - b^2}{3a^2}\right)y + \left(\frac{2b^3 - 9abc + 27a^2d}{27a^3}\right) = 0</math>. | ||
− | <math>u</math> and <math>v</math> are the roots of the | + | <math>u</math> and <math>v</math> are the roots of the system of equations <math>\begin{cases} uv = \frac{p^3}{27} \\ v - u = q \end{cases}</math>. We can solve this by substitution: |
<math>v = q + u</math> (We are still using p and q because they might get a little messy if we use p and q in terms of a, b, c, and d.) | <math>v = q + u</math> (We are still using p and q because they might get a little messy if we use p and q in terms of a, b, c, and d.) | ||
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<math>- \sqrt[3]{\frac{\left(\frac{2b^3 - 9abc + 27a^2d}{27a^3}\right) \pm \sqrt{\frac{3\left(\frac{2b^3 - 9abc + 27a^2d}{9a^3}\right)^2 - 4\left(\frac{3ac - b^2}{3a^2}\right)^3\left(\frac{2b^3 - 9abc + 27a^2d}{27a^3}\right)}{27}}}{2}} - \frac{b}{3a}</math> | <math>- \sqrt[3]{\frac{\left(\frac{2b^3 - 9abc + 27a^2d}{27a^3}\right) \pm \sqrt{\frac{3\left(\frac{2b^3 - 9abc + 27a^2d}{9a^3}\right)^2 - 4\left(\frac{3ac - b^2}{3a^2}\right)^3\left(\frac{2b^3 - 9abc + 27a^2d}{27a^3}\right)}{27}}}{2}} - \frac{b}{3a}</math> | ||
− | (See? I told you it would be messy.) I'm not going to simplify all that squaring and cubing right now: maybe soon! | + | (See? I told you it would be messy.) I'm not going to simplify all that squaring and cubing right now: maybe soon! Also, if you select <math>+</math> for the first equation and <math>-</math> for the second (or vice versa--they lead to the same number), you will always get a real number. |
+ | |||
+ | One last piece of advice: '''''Don't try to memorize this.''''' Memorize the process (shortcut: just look at TLDR for each section). [[Cubic polynomial|Here]] is another way to do it. | ||
===If you're just asking for the formula for a monic cubic...=== | ===If you're just asking for the formula for a monic cubic...=== | ||
− | Here is the formula: | + | Here is the formula for <math>x^3 + ax^2 + bx + c</math>: |
− | <math>x = \sqrt[3]{\frac{-\left(\frac{2a^3 - 9ab + | + | <math>x = \sqrt[3]{\frac{-\left(\frac{2a^3 - 9ab + 27c}{27}\right) \pm \sqrt{\frac{3\left(\frac{2a^3 - 9ab + 27c}{9}\right)^2 - 4\left(\frac{3b - a^2}{3}\right)^3\left(\frac{2a^3 - 9ab + 27c}{27}\right)}{27}}}{2}} - \sqrt[3]{\frac{\left(\frac{2a^3 - 9ab + 27c}{27}\right) \pm \sqrt{\frac{3\left(\frac{2a^3 - 9ab + 27c}{9}\right)^2 - 4\left(\frac{3b - a^2}{3}\right)^3\left(\frac{2a^3 - 9ab + 27c}{27}\right)}{27}}}{2}} - \frac{a}{3}</math> |
+ | |||
+ | ===If you're asking for the formula for a depressed monic cubic...=== | ||
+ | |||
+ | Here is the formula for <math>x^3 + ax + b</math>: | ||
+ | <math>x = \sqrt[3]{\frac{-b \pm \sqrt{\frac{27b^2 - 4a^3b}{27}}}{2}} - \sqrt[3]{\frac{b \pm \sqrt{\frac{27b^2 - 4a^3b}{27}}}{2}}</math> |
Revision as of 14:42, 11 December 2020
A cubic equation is an equation of the form:
.
A cubic equation has 3 roots, either all real OR one real, two complex.
Contents
Solving Cubic Equations
If you're too lazy to follow, look at subsection "TLDR" for each section.
Converting to a Depressed Equation
You start with the equation .
Divide both sides by a: .
Now we change the coefficient of to
(e.g. change it to a depressed cubic). We do this by substituting
, giving:
.
is
and
is
, so now we have
.
TLDR?
The equation is where
and
.
Solving the Depressed Equation
Now here comes the smart part. Substitute .
The equation becomes . Simplification:
We want that last term to equal , so we can set
. (We can't use
, because then
, which is not necessarily true.) Solving this equation gives us
. If
, then
. We now have a system of equations:
.
We can solve this via the quadratic formula. After and
are obtained, we have
and
.
TLDR?
where u and v are roots of the system
.
The Cubic Formula
The cubic formula can be obtained by using the above method. These are the steps:
The depressed cubic is of the form .
and
are the roots of the system of equations
. We can solve this by substitution:
(We are still using p and q because they might get a little messy if we use p and q in terms of a, b, c, and d.)
(comes from
)
(See? I told you it would be messy.) I'm not going to simplify all that squaring and cubing right now: maybe soon! Also, if you select for the first equation and
for the second (or vice versa--they lead to the same number), you will always get a real number.
One last piece of advice: Don't try to memorize this. Memorize the process (shortcut: just look at TLDR for each section). Here is another way to do it.
If you're just asking for the formula for a monic cubic...
Here is the formula for :
If you're asking for the formula for a depressed monic cubic...
Here is the formula for :