Basic functions: Polynomial functions
Quadratic equations in disguise
Sometimes you can convert an equation that seemingly has nothing to do with quadratic equations by a trick into a quadratic equation. Some examples illustrate tricks like substitution, squaring, distinguishing cases, and reduction.
\(x=\sqrt{6}\quad\lor\quad x=-\sqrt{6}\)
\(x^4-3x^2-18=0\) is a fourth degree polynomial equation, but setting \(y=x^2\), it becomes a quadratic equation in \(y\) : \[y^2-3y-18=0\] Factorisation by inspection leads to the following equation in \(y\): \[(y-6)(y+3)=0\] with solutions \[y=6\quad\vee\quad y=-3\] But because \(y=x^2\), and a square of a real number cannot be negative, the equation\(y=-3\) does not lead to solutions. What remains is the equation \(x^2=6\) with two solutions: \[x=\pm \sqrt{6}\]
\(x^4-3x^2-18=0\) is a fourth degree polynomial equation, but setting \(y=x^2\), it becomes a quadratic equation in \(y\) : \[y^2-3y-18=0\] Factorisation by inspection leads to the following equation in \(y\): \[(y-6)(y+3)=0\] with solutions \[y=6\quad\vee\quad y=-3\] But because \(y=x^2\), and a square of a real number cannot be negative, the equation\(y=-3\) does not lead to solutions. What remains is the equation \(x^2=6\) with two solutions: \[x=\pm \sqrt{6}\]
