### 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.

\(x=1\quad\lor\quad x=-1\)

\(x^4+3x^2-4=0\) is a fourth degree polynomial equation, but setting \(y=x^2\), it becomes a quadratic equation in \(y\) : \[y^2+3y-4=0\] Factorisation by inspection leads to the following equation in \(y\): \[(y-1)(y+4)=0\] with solutions \[y=1\quad\vee\quad y=-4\] But because \(y=x^2\), and a square of a real number cannot be negative, the equation\(y=-4\) does not lead to solutions. What remains is the equation \(x^2=1\) with two solutions: \[x=\pm 1\]

\(x^4+3x^2-4=0\) is a fourth degree polynomial equation, but setting \(y=x^2\), it becomes a quadratic equation in \(y\) : \[y^2+3y-4=0\] Factorisation by inspection leads to the following equation in \(y\): \[(y-1)(y+4)=0\] with solutions \[y=1\quad\vee\quad y=-4\] But because \(y=x^2\), and a square of a real number cannot be negative, the equation\(y=-4\) does not lead to solutions. What remains is the equation \(x^2=1\) with two solutions: \[x=\pm 1\]

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