Quadratic Equations
6. Equations Reducible to Quadratic Form
6.3. Form 3
Equation of the form \(a{x^{\bf{4}}} + b{x^{\bf{3}}} + c{x^{\bf{2}}} + bx + a = {\rm{ }}{\bf{0}}\). In such cases divide the equation by \(x^2\) and then assume \(x + \frac{1}{x}=t\). See the example.
Example 01: Solve the equations
\(a{x^4} + {\rm{ }}b{x^3} + {\rm{ }}c{x^3} + {\rm{ }}bx{\rm{ }} + {\rm{ }}a{\rm{ }} = 0\)
Solution: this equations is also convertible to quadratic form, dividing by \({x^2}\), we get,
\(a{x^2} + bx + c + \frac{b}{x} + \frac{a}{{{x^2}}} = 0\)
\(a\left( {{x^2} + \frac{1}{{{x^2}}}} \right) + b\left( {x + \frac{1}{x}} \right) + c = 0\)
Suppose \(\left( {x + \frac{1}{x}} \right)\) = t, then \(\left( {{x^2} + \frac{1}{{{x^2}}}} \right)\) \( = {t^2}-{\rm{ }}2\)
Then the equation becomes,
\(a\left( {{t^2}-{\rm{ }}2} \right){\rm{ }} + b\left( t \right){\rm{ }} + c = {\rm{ }}0\), this equation can be solved as discussed in the previous example.