Quadratic equations in disguise

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.

Find the exact real solutions of the following equation.

$x^4-16=0$

$x=2\quad\lor\quad x=-2$

$x^4-16=0$ is a fourth degree polynomial equation, but setting $y=x^2$ , it becomes a quadratic equation in $y$ :

$y^2-16=0$ Factorisation by inspection leads to the following equation in $y$ :

$(y-4)(y+4)=0$ with solutions

$y=4\quad\vee\quad y=-4$ 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 $x^2=4$ with two solutions:

$x=\pm 2$

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