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.
\(x=1\quad\lor\quad x=-1\)
\(x^4+5x^2-6=0\) is a fourth degree polynomial equation, but setting \(y=x^2\), it becomes a quadratic equation in \(y\) : \[y^2+5y-6=0\] Factorisation by inspection leads to the following equation in \(y\): \[(y-1)(y+6)=0\] with solutions \[y=1\quad\vee\quad y=-6\] But because \(y=x^2\), and a square of a real number cannot be negative, the equation\(y=-6\) does not lead to solutions. What remains is the equation \(x^2=1\) with two solutions: \[x=\pm 1\]
\(x^4+5x^2-6=0\) is a fourth degree polynomial equation, but setting \(y=x^2\), it becomes a quadratic equation in \(y\) : \[y^2+5y-6=0\] Factorisation by inspection leads to the following equation in \(y\): \[(y-1)(y+6)=0\] with solutions \[y=1\quad\vee\quad y=-6\] But because \(y=x^2\), and a square of a real number cannot be negative, the equation\(y=-6\) does not lead to solutions. What remains is the equation \(x^2=1\) with two solutions: \[x=\pm 1\]
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