Solving quadratic equations and inequalities: Quadratic equations
Quadratic equations in disguise
Sometimes you can turn an equation that seeming has nothing to do with quadratic equations by a trick into a quadratic equation. Some examples illustrate tricks like substitution, squaring, distinguishing case, and reduction.
\(x^4-2x^2-3=0\) is a fourth degree polynomial equation, but setting \(y=x^2\), it becomes a quadratic equation in \(y\) : \[y^2-2y-3=0\] Factorisation by inspection leads to the following equation in \(y\): \[(y-3)(y+1)=0\] with solutions \[y=3\quad\vee\quad y=-1\] But because \(y=x^2\), and a square of a real number cannot be negative, the equation\(y=-1\) does not lead to solutions. What remains is the equation \(x^2=3\) with two solutions: \[x=\pm \sqrt{3}\]
