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.
\(x=\sqrt{3}\quad\lor\quad x=-\sqrt{3}\)
\(x^4-x^2-6=0\) is a fourth degree polynomial equation, but setting \(y=x^2\), it becomes a quadratic equation in \(y\) : \[y^2-y-6=0\] Factorisation by inspection leads to the following equation in \(y\): \[(y-3)(y+2)=0\] with solutions \[y=3\quad\vee\quad y=-2\] But because \(y=x^2\), and a square of a real number cannot be negative, the equation\(y=-2\) does not lead to solutions. What remains is the equation \(x^2=3\) with two solutions: \[x=\pm \sqrt{3}\]
\(x^4-x^2-6=0\) is a fourth degree polynomial equation, but setting \(y=x^2\), it becomes a quadratic equation in \(y\) : \[y^2-y-6=0\] Factorisation by inspection leads to the following equation in \(y\): \[(y-3)(y+2)=0\] with solutions \[y=3\quad\vee\quad y=-2\] But because \(y=x^2\), and a square of a real number cannot be negative, the equation\(y=-2\) does not lead to solutions. What remains is the equation \(x^2=3\) with two solutions: \[x=\pm \sqrt{3}\]
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