Quadratic equations in disguise

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.

Find the exact real solutions of the following equation.

$x^4-4x^2-5=0$

$x=\sqrt{5}\quad\lor\quad x=-\sqrt{5}$

$x^4-4x^2-5=0$ is a fourth degree polynomial equation, but setting $y=x^2$ , it becomes a quadratic equation in $y$ :

$y^2-4y-5=0$ Factorisation by inspection leads to the following equation in $y$ :

$(y-5)(y+1)=0$ with solutions

$y=5\quad\vee\quad y=-1$ 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 $x^2=5$ with two solutions:

$x=\pm \sqrt{5}$

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