One-Variable Higher-Order Equations for Physics Help

Introduction

As the exponents in single-variable equations become larger and larger, finding the solutions becomes an ever more complicated and difficult business. In the olden days, a lot of insight, guesswork, and tedium were involved in solving such equations. Today, scientists have the help of computers, and when problems are encountered containing equations with variables raised to large powers, brute force is the method of choice. We’ll define cubic equations, quartic equations, quintic equations , and nth-order equations here but leave the solution processes to the more advanced pure-mathematics textbooks.

The Cubic

A cubic equation , also called a one-variable third-order equation or a third-order equation in one variable , can be written in the following standard form:

ax ³ + bx ² + cx + d = 0

where a, b, c , and d are constants, and x is the variable. (Here, c does not stand for the speed of light in free space but represents a general constant.) If you’re lucky, you’ll be able to reduce such an equation to factored form to find real-number solutions r, s , and t:

( x − r ) ( x − s ) ( x − t ) = 0

Don’t count on being able to factor a cubic equation into this form. Sometimes it’s easy, but usually it is exceedingly difficult and time-consuming.

The Quartic

A quartic equation , also called a one-variable fourth-order equation or a fourth-order equation in one variable , can be written in the following standard form:

ax ⁴ + bx ³ + cx ² + dx + e = 0

where a, b, c, d , and e are constants, and x is the variable. (Here, c does not stand for the speed of light in free space, and e does not stand for the exponential base; instead, these letters represent general constants in this context.) There is an outside chance that you’ll be able to reduce such an equation to factored form to find real-number solutions r, s, t , and u:

( x − r ) ( x − s ) ( x − t ) ( x − u ) = 0

As is the case with the cubic, you will be lucky if you can factor a quartic equation into this form and thus find four real-number solutions with ease.

The Quintic

A quintic equation , also called a one-variable fifth-order equation or a fifth-order equation in one variable , can be written in the following standard form:

ax ⁵ + bx ⁴ + cx ³ + dx ² + ex + f = 0

where a, b, c, d, e , and f are constants, and x is the variable. (Here, c does not stand for the speed of light in free space, and e does not stand for the exponential base; instead, these letters represent general constants in this context.) There is a remote possibility that if you come across a quintic, you’ll be able to reduce it to factored form to find real-number solutions r, s, t, u , and v: 

( x − r ) ( x − s ) ( x − t ) ( x − u ) ( x − v ) = 0

As is the case with the cubic and the quartic, you will be lucky if you can factor a quintic equation into this form. The “luck coefficient” goes down considerably with each single-number exponent increase.

The N Th-order Equation

A one-variable n th-order equation can be written in the following standard form:

a ₁ x ⁿ + a ₂ x ^{n −1} + a ₃ x ^{n −2} + ... + a _{n −2} x ² + a _{n −1} x + a _n = 0

where a ₁ , a ₂ ,..., a _n are constants, and x is the variable. We won’t even think about trying to factor an equation like this in general, although specific cases may lend themselves to factorization. Solving equations like this requires the use of a computer or else a masochistic attitude.

Practice problems for these concepts can be found at:  Equations, Formulas, And Vectors for Physics Practice Test